Evolutionary Developmental Soft Robotics: Towards Adaptive and Intelligent Machines Following Nature's Approach to Design

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SANT’ANNA SCHOOL OF ADVANCED STUDIES

The BioRobotics Institute PhD in BioRobotics – XXIX Cycle

PhD Thesis

Evolutionary Developmental Soft

Robotics:

Towards Adaptive and Intelligent Machines

Following Nature’s Approach to Design

Francesco Corucci

Tutor Supervisor

Prof. Josh Bongard

Co-tutor:

Dr. Matteo Cianchetti Prof. Cecilia Laschi

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One of the dreams of researchers in fields such as bionics and artificial intelligence is to be able one day to build adaptive machines approaching the complexity and sophistication of biological creatures, both in terms of life-like morphology and behavior. To this end roboticists have designed increasingly complex machines following a mechatronic approach, later realizing that it was indeed extremely difficult to have them solving even the simplest tasks, such as walking, or grasping an object. Simultaneously, artificial intelligence researchers have traditionally focused their attention on high level cognition, abstract symbol processing, and neural computation, later coming to the conclusion that intelligence starts, in fact, with the body and its dynamic interaction with the surrounding environment. The realization that morphology, and, particularly, a ”soft” one, may play a fundamental role in the emergence of intelligence and adaptive behavior has led to research fields such as embodied intelligence and soft robotics, whose premises are to revolutionize both the traditional mechatronic approach to robotics, as well as the study of biological and artificial intelligence. Soft robots are built out of all sorts of compliant and smart materials, that will soon integrate distributed sensing and actuation, as well as unprecedented capabilities that will allow them to grow, change shape and mechanical properties, self-heal, gradually bridging the existing complexity gap among living beings and machines. Nevertheless, soft robotics is also bringing countless problems to the table, related to design, sensing, control and fabrication methodologies. As for the study of intelligence, despite many intuitions, the relationship between softness and intelligence and the conditions under which adaptive and intelligent behavior might emerge in a soft-bodied creature are, to date, still largely unclear. If we look at Nature, however, there is one fundamental, conceptual aspect which motivates the research presented in this thesis, which is the following: biological complexity has emerged from bottom-up, self-organizing, adaptive processes, such as evolution and development. Researchers and engineers tend to approach issues from a top-down perspective instead. They break down problems, they build machines starting from high-level requirements: this approach does not always work when the goal is to study or replicate biological phenomena. We try to understand what intelligence is, how it can be quantified, we struggle trying to understand how the brain works. In the midst of countless models and theories, there is one simple truth we can rely on: brains, bodies, intelligence, adaptive strategies, have all emerged from self-organizing adaptive processes, such as, again, evolution and development. In the spirit of bio-inspiration, i.e. of taking inspiration from Nature in order to devise new engineering solutions, it thus makes sense to replicate natural processes, instead of their products. With all this in mind, in this thesis we present a number of results in the field of evolutionary developmental soft robotics (evo-devo-soro, for brevity), a broad research area drawing ideas from and contributing to a number of

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employ biologically-inspired computational processes in order to design robots, which can be characterized by complex dynamic morphologies composed of multiple materials, including compliant ones. While providing several possibilities for simulation studies targeting phenomena involved in the emergence of intelligent and adaptive behavior, these tools represent at the same time extremely powerful design automation techniques, which may soon allow the automated design and fabrication of complete, self-sufficient, optimized, adaptive machines for different tasks and environments. Reported in order of increasing technical and conceptual complexity, different application scenarios of these techniques will be presented in this work, ranging from engineering applications of evolutionary algorithms in the first chapters, to the implementation of very general evolutionary and developmental system in the last ones. These more general tools are exploited in order to perform extensive simulation studies which attempt to shed light on various phenomena involved in the emergence of adaptive and intelligent behavior, such as the effects of material properties on the evolution of soft locomotion and morphological computation, or the evolution adaptive laws of morphological developmental plasticity in soft machines, and their implications in terms of robustness and resilience. Taking the bottom-up approach to its extreme consequences, and with the aim of isolating different phenomena and effects, our investigations will focus on morphology evolution, with control being kept deliberately simple. Overall, with this work we hope to bring new evidences in fields such as soft robotics and embodied intelligence, which may help unleash the full potential of these disciplines.

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Publications list

Journal publications

• F. Corucci, N. Cheney, S. Kriegman, F. Giorgio-Serchi, J. Bongard, C. Laschi, ”Evolving soft locomotion in aquatic and terrestrial environments: effects of material

properties and environmental transitions”, Soft Robotics (under review)

• F. Corucci, N. Cheney, S. Kriegman, J. Bongard, C. Laschi, ”Evolutionary developmental soft robotics as a framework to study intelligence and adaptive behavior in animals and plants”, submitted to Frontiers in Robotics and AI (under review)

• F. Giorgio-Serchi, A. Arienti, F. Corucci, M. Giorelli, C. Laschi, ”Hybrid pa-rameter identification of a multi-modal underwater soft robot”, Bioinspiration & Biomimetics 12.2 (2017): 025007.

• M. Calisti, F. Corucci, A. Arienti, C. Laschi, ”Dynamics of underwater legged lo-comotion: modeling and experiments on an octopus-inspired robot”, Bioinspiration & Biomimetics 10.4 (2015): 046012. Featured Article

• M. Calisti, M. Cianchetti, M. Manti, F. Corucci, C. Laschi, ”Contest-driven soft-robotics boost: the RoboSoft Grand Challenge”, In: Frontiers Robotics and AI, (2016), 3, 55.

Book chapters

• F. Corucci, ”Evolutionary Developmental Soft Robotics: towards adaptive and intelligent soft machines following nature’s approach to design”, In Soft Robotics: Trends, Applications and Challenges, Springer, 2017, 111-116

• H. Hauser, F. Corucci, ”Morphosis – Taking Morphological Computation to the Next Level”, In Soft Robotics: Trends, Applications and Challenges, Springer, 2017, 117-122

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• S. Kriegman, N. Cheney, F. Corucci, J. Bongard, ”A Minimal Developmental Model Can Increase Evolvability in Soft Robots”, In: Proceedings of the Genetic and Evolutionary Computation Conference (GECCO), 2017 (accepted, to appear) • F. Corucci, N. Cheney, H. Lipson, C. Laschi, and J. Bongard, ”Material properties affect evolution’s ability to exploit morphological computation in growing soft-bodied creatures”, In: Proceedings of ALIFE XV, The Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 2016

• F. Corucci, M. Calisti, H. Hauser, C. Laschi, ”Evolutionary discovery of self-stabilized dynamic gaits for a soft underwater legged robot”, In: Proceedings of the 17th IEEE International Conference on Advanced Robotics (ICAR2015), 2015 • F. Corucci, M. Calisti, H. Hauser, C. Laschi, ”Novelty-based evolutionary design

of morphing underwater robots”, In: Proceedings of the Genetic and Evolutionary Computation Conference (GECCO), 2015 - Best paper award nomination

• M. Calisti, F. Corucci, A. Arienti, C. Laschi, ”Bipedal walking of an octopus-inspired robot”, In: Biomimetic and Biohybrid Systems - Living Machines 2014, Springer Lectures Notes in Artificial Intelligence, 2014

• V. Cacucciolo*, F. Corucci*, M. Cianchetti, C. Laschi, ”Evolving optimal swim-ming in different fluids: a study inspired by batoid fishes”, In: Biomimetic and Biohybrid Systems - Living Machines 2014, Springer Lectures Notes in Artificial Intelligence, 2014 (* equal contribution)

Workshops, abstracts and posters

• Kriegman, S., Cappelle, C., Corucci, F., Bernatskiy, A., Cheney, N., Bongard, J., ”Simulating the Evolution of Soft and Rigid-Body Robots”, Workshop on Simula-tion in EvoluSimula-tionary Robotics, Genetic and EvoluSimula-tionary ComputaSimula-tion Conference (GECCO), 2017 (under review)

• F. Corucci, N. Cheney, H. Lipson, C. Laschi, and J. Bongard, ”Evolving swimming soft-bodied creatures”, In ALIFE XV, The Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 2016 - Second best poster at ALIFE2016 (> 60 entries), finalist in the 2016 GECCO Virtual Creatures Contest (Honorable Mention for Aesthetic Appeal)

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a sustainable and adaptive treehouses ecosystem in the Chilean forest,” In ALIFE XV, The Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 2016

• F. Corucci, M. Calisti, H. Hauser, C. Laschi ”Shaping the body to shape the behavior: a more active role of the morphology in the brain-body trade-off”, 13th European Conference on Artificial Life (ECAL 2015), 2015

• F. Corucci, ”Evolutionary design of morphing underwater robots”, Soft Robotics Week 2015, Livorno, Italy, 2015

• V. Cacucciolo*, F. Corucci*, M. Cianchetti, C. Laschi, ”Evolving optimal swim-ming in different fluids: a study inspired by batoid fishes”, 1st RoboSoft Plenary Meeting, Pisa, Italy, 2014 (* equal contribution)

• F. Corucci, M. Calisti, A. Arienti, C. Laschi, ”The locomotion of an underwater soft robot as a case study for further investigating morphological computation and embodied intelligence”, 1st RoboSoft Plenary Meeting, Pisa, Italy, 2014

• V. Cacucciolo*, F. Corucci*, M. Cianchetti, C. Laschi, ”Evolving optimal swim-ming behavior in different fluids: a study inspired by batoid fishes”, Italian National Bioengineering Group, National Congress, 2014 (* equal contribution)

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I would like to thank the following people and institutions:

• The BioRobotics Institute of the Sant’Anna School of Advanced Studies (Pisa, Italy) and Prof. Cecilia Laschi, which have supported my research and

gave me the freedom to pursue my scientific interests.

• The Research Centre on Marine Robotics and Sea Technologies (Livorno, Italy), where I have developed part of my research activities. Among the people I have met there, I would like to thank Andrea Arienti especially, a true friend who has helped me overcoming many difficult moments. Thanks to Francesco Giorgio-Serchi and Federico Renda as well for the many chats and discussions.

• Dr. Helmut Hauser, for several intellectual exchanges and for helping me

establishing important professional collaborations, that resulted to be fundamental for my research.

• Prof. Josh Bongard for welcoming and advising me during my one-year stay at the Morphology, Evolution & Cognition Laboratory (MECL), Vermont Complex Systems Center, University of Vermont (Burlington, VT, USA). • All the friends I have met in the United States (Roman, Mila, Sam, Nick, Collin,

Anton, Chris, ...), who have contributed to making my stay more pleasant and fun. Thanks especially to Marcin and Michalina Szubert, my fun polish flatmates, for the time spent together in Vermont, and for being kind enough to help me and my girlfriend getting settled in Burlington after a pretty rough start.

• In 2016 alone, my evolutionary simulations required more than five millions dedi-cated CPU hours: the equivalent of almost 600 years of computation. Thanks to the Vermont Advanced Computing Core for providing these computational resources, and to Andrew Evans, Jim Lawson and Andrea Elledge for maintaining the cluster.

• The external reviewers of this PhD Thesis, and the PhD committee.

• My girlfriend Chiara and my family, for bearing with me during all these past years of study and difficult decisions.

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• RoboSoft - A Coordination Action for Soft Robotics (FP7-ICT-2013-C # 619319) • Human Brain Project (FP7/2007-2013 # 604102)

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Introduction

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Despite many recent successes in robotics and artificial intelligence, robots are still far from matching the performances of biological creatures outside controlled environments. We argue that this is mainly due to their lack of adaptivity. By taking inspiration from Nature, biologically-inspired robotics (or biorobotics) and soft robotics have pointed out new directions towards this goal, showing a lot of potential. However, in many ways, this potential is still largely unexpressed. Three main limiting factors can be identified: 1) the common adoption of non-scalable, top-down design processes constrained by human capabilities, 2) an excessive focus on proximal solutions observed in Nature instead of on the natural processes that gave rise to them, 3) the lack of general insights regarding intelligence, adaptive behavior, and the conditions under which they emerge in Nature. By adopting algorithms inspired by the natural evolution and development, evolutionary developmental soft robotics — the main unifying topic of this thesis — represents in a way the ultimate form of bio-inspiration. This approach allows the automated design of complete soft robots, whose morphology, control, and sensory system are co-optimized for different tasks and environments, and can adapt in response to environmental stimuli during their lifetime. In this thesis a number of studies supported by this type of methodologies will be reported, ranging from the evolutionary design of real soft robots for practical applications, to the study of general properties of soft bodied creatures. The goals of this work can be summarized in three main points: 1) to help robotics and soft robotics overcome some of their technical limitations (such as the difficulty

1_{Partly based on:}

• F. Corucci, Evolutionary developmental soft robotics: Towards adaptive and intelligent soft machines following nature’s approach to design. In Soft Robotics: Trends, Applications and Challenges, pages 111–116. Springer International Publishing, 2017.

• F. Corucci, N. Cheney, S. Kriegman, J. Bongard, C. Laschi, Evolutionary developmental soft robotics as a framework to study intelligence and adaptive behavior in animals and plants (submitted to Frontiers in Robotics and AI, under review)

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to cope with increasingly large design spaces); 2) to study phenomena involved in the evolution and emergence of adaptive and intelligent behavior (such as embodied cognition and morphological computation), which will be key to the widespread of robots outside controlled environments; 3) to help robotics and soft robotics realizing their full potential for adaptivity by incorporating adaptive processes borrowed from Nature into their design processes (i.e. evolutionary and developmental dynamics).

1.1 A different take on bioinspiration

Researchers in the field of biologically-inspired robotics [1, 2] take inspiration from the natural world in order to distill effective mechanism observed in animals [3] and plants [4–7] into new engineering solutions. The goal is not just to imitate Nature, but rather to extract general principles underlying the evolutionary success of biological creatures, later applying these insights to the development of more effective and efficient robotic systems. Although this procedure entails a number of positive effects (e.g. a deeper comprehension of biology – ”understanding by building” [8, 9], the possibility to overcome technological barriers by finding alternative solutions to complex problems, etc.), there is an aspect of conventional bio-inspired design that deserves to be highlighted: researchers in this field take inspiration from the products of a number of natural processes (most notably, evolution) that occur in the natural world. This has a number of important consequences that are worth examining, that could be in some cases misleading depending on the goal one is interested in reaching when applying bioinspiration.

First, the creatures that populate and have populated our world are the result of a single evolutionary trajectory, determined by continuous incremental adaptations that allowed them to survive in a changing environment, facing the opponents that happened to compete for the same resources at a given time, in the same ecological niche [10, 11]. Therefore, they are not necessarily optimal in general terms, as we would like machines to be in the engineering field. We may be interested in understanding and replicating a specific behavior or morphological trait observed in an animal, but nothing guarantees that that behavior/trait is the best solution or source of inspiration for a given task we may want a robot to solve: it is simply one of the possible solutions Nature has found. It could have discovered a different, maybe better one, under different circumstances. Secondly, animals and plants are constrained by the biological substrate that was available on Earth. Biological bodies are made of cells, which have specific properties and require specific conditions to survive. At a coarser grain, biological brains are composed by neurons, which have specific characteristics. In robotics and artificial intelligence, however, we do not have the same constraints Nature has. Humans have invented

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sophisticated technologies (e.g. robust and lightweight materials, high-speed computation and communication technologies, all sorts of sensors and actuators, etc.) that cannot be found in Nature, and could not be, therefore, exploited during natural evolution. It can be expected that different building blocks and different constraints will result in optimal solutions being potentially very different in the artificial and natural realms. On the other hand, biological inspiration pushes us towards looking for a convergence between these two worlds.

Third, animals and plants evolved to cope with challenges such as survival, foraging and reproduction. Although a parallelism can be made among these tasks and those that robots are (or will be) required to accomplish (e.g. surviving = avoid being destroyed in a hostile environment, foraging = harvest energy and resources in order to self-sustain, etc.), robots are not in general required to solve tasks that are of paramount importance in the natural world (e.g. they do not need to mate and reproduce), which shaped biological creatures.

Fourth, although bioinspired robotics can help unveiling specific aspects of animal and plant intelligence, more general tools may be needed in order to gain a deeper understanding and ability to implement general forms of intelligent and adaptive behavior. To date, the conditions under which such behavior emerges (or can emerge) in plants, animals and machines are still largely unknown.

One fundamental point, however, is the fact that biological intelligence has been produced by an evolutionary process. These observations entail a paradigm shift in bio-inspiration: despite the solutions that we observe in the natural world being clever and astonishing, what is really special about them is, more than each specific mechanisms, the processes that produced them, and the fact that they managed to do so with neither guidance, nor goals.

This justifies fields such as evolutionary robotics [12–15] and developmental robotics [16] (and, more generally, artificial life [17]), in which inspiration is taken from biological processes, that are generalized and instantiated into artificial systems (”in silico”). The emergent properties of the results of these (now artificial) processes can be then analyzed, with the additional advantage of having free access to the complete state of the system (the same cannot be said when studying living beings).

Replicating processes instead of products also allows moving from the study of ”life-as-we-know-it” to that of ”life-as-it-could-be” [17], under different circumstances with respect to the ones we have had in biological evolution. In a sense, following

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the aforementioned considerations on bio-inspiration, this is exactly what we may be interested in: observing what robots would (and should) look like given our technological substrate and evolutionary and developmental processes similar to those that shaped living beings.

This approach gives us a remarkable freedom, too. Once one of such processes has been abstracted in artificial form (in the form of an algorithm), it can be instantiated in many different ways, simulating different starting points, environmental conditions, building blocks that can be manipulated, etc. From an artificial intelligence perspective, the idea is that the possibility to manipulate artificial evolutionary and developmental processes, to simulate many different task environments and to have access to all aspects of the evolving creatures (genotype, phenotype, sensory information, etc.) will enable a deeper understanding of the basic phenomena underlying adaptive and intelligent behavior. These kinds of simulation studies also allow to detach from current technological constraints, developing theories and concepts that can inform neighboring fields. Finally, in addition to supporting hypothesis testing, these techniques effectively constitute powerful design automation tools [18], allowing to devise a wide array of optimized designs for different task environments, that may be soon automatically fabricated [19–21] and deployed [22–24].

In what follows some examples will be described in which algorithms inspired by natural evolution (Sect. 1.4) are applied to the design of soft robots (Sect. 1.2), combined, in some cases, with developmental paradigms. This research area can be broadly referred to as evolutionary developmental soft robotics (evo-devo-soro) [25, 26] (Sect. 1.7), a subfield laying at the intersection between artificial life and robotics, in which complete soft robots —both their morphology, sensory and control systems— can be automatically evolved and grown instead of being manually designed. As we will point out, this can be done with several different purposes in mind: from the design and optimization of real robots for specific tasks, to the study of general properties of soft bodied creatures.

1.2 Soft robotics

Soft robotics [27] is nowadays a very popular research field: despite booming approxi-mately in the last ten years, early demonstrations of soft robots stretch back to the early nineties [28]. There are many alternative definition of what a soft robot is (or should be) [27, 29–32]. From now on, we will refer to the following, that was worked out in the context of the RoboSoft CA Project (a Coordination Action for Soft Robotics funded by the European Commission) [33]: soft robots are ”robots/devices that can actively

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Figure 1.1: Soft robots examples (reprinted from: [27]). Credits (left to right): iSprawl, M. Cutkosky; X-RHex, K. C. Galloway; soft robotic fish, D. Rus; OCTOPUS robot, photographer J. Hills; PoseiDRONE robot, photographer M. Brega; universal gripper, 2012 IEEE; Origami robot, J. K. Paik; Softworm, B. A. Trimmer; soft robotic glove, P.

Polygerinos; inflatable robot, 2012 IEEE; Octobot, J. A. Lewis

interact with the environment and can undergo large deformations relying on inherent or structural compliance”. This usually entails that this kind of robots are built out of flexible materials (e.g. different types of rubber-like materials such as silicone), totally or in part (Fig. 1.1). As a consequence, traditional technologies for actuation, sensing, modeling and control need to be revised, or even reinvented from scratch based on completely new principles.

For the sake of completeness, it is worth noting that there exist different interpreta-tions and even completely orthogonal approaches to soft robotics with respect to the aforementioned definition. In different contexts [34], for example, in order to qualify as ”soft” a robot does not need to be built out of deformable materials, but rather it should exhibit a soft interaction with the environment. This can be realized even in robots featuring rigid links and electrical motors, by applying sophisticated control strategies (e.g. compliance or impedence control of variable stiffness joints). These approaches are,

however, outside the scope of this work.

For our purposes, softness is simply intended as the possibility of embedding deformable materials, sensors, and/or actuators into the robots (or virtual creatures) at hand, in addition to or in substitution of more traditional rigid links and electrical motors as traditionally found in robotics. Also to be noted that softness will not usually be imposed from a top-down perspective. In general robots will have the possibility of being soft, to different extents, completely or just in part, only if there is an advantage in incorporating compliant elements into the design for a given task and environment. More often than not, the material properties of the robots at hand will represent one or more variables to be controlled, manipulated either by the experimenter, or by an algorithm.

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But why should one bother dealing with this class of ”exotic” robots, bearing with all the additional difficulties that this entails in terms of modeling, design, fabrication, sensing, and control?

First of all, the fact that softness is found in many biological systems [30] makes it interesting already from a biorobotics perspective. Present to different extents in a wide array of creatures (often in combination with rigid subsystems, e.g. skeletal structures), several studies [30] have focused on how softness contributes to the success of different biological creatures. From an engineering standpoint, instead, it has been shown how robots embedding compliant materials can passively or actively conform to objects (useful, for example, for manipulation tasks) [35, 36] or to surfaces [37] (e.g. during locomotion), which can result in an intrinsically safer and more adaptive interaction with humans [38] and with the environment [37]. Their ability to change shape and to reconfigure can allow them to reach places that rigid robots cannot access: they can navigate narrow spaces [39], or squeeze into small apertures [40]. Soft robots also allow for virtually infinite degrees of freedom, which make them particularly interesting in the context of robotic manipulation [41]. Moreover, with progresses in material sciences, they promise to realize advanced and life-like capabilities [42] that are beyond those of their rigid counterparts, such as self-healing [43, 44]. Lastly, the use of soft materials allows in many cases to dramatically reduce the cost associated with the fabrication of these machines: with progresses in 3D printing and soft fabrication [19] it could be soon possible to fabricate large quantities of cheap, self-contained, resilient soft robots, e.g. for applications such as search and rescue: this is particularly promising especially in light of artificial intelligence techniques such as swarm intelligence [45], that allow to coordinate the operation of large groups of distributed robots.

All this comes at a cost, however, in the form of both technological and conceptual limitations. From the technological point of view, there are still many challenges to overcome when it comes to fabricating reliable, complete soft robots, integrating flexible and distributed sensors, actuators, and power supply. From a conceptual point of view, most of the consolidated methodologies for the design and control of conventional robotics systems cannot be applied to soft machines, and need to be reinvented. Finally, it is to be noted how the design space of soft machines is considerably bigger and complex than that of their rigid counterparts, including lots of new parameters associated to their material properties, as well as to their distributed sensing and control systems: progresses in fields such as smart materials, flexible electronics and, ultimately, nanotechnologies, are not going to ameliorate this situation, with possibly countless new design dimensions to be taken into account. This poses considerable challenges, but opens at the same time several possibilities for innovation [46].

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1.3 Artificial Intelligence, Embodied Cognition,

Morpho-logical Computation

Soft robotics is also extremely interesting from the point of view of Artificial Intelligence (AI). Traditionally focused on the brain, on abstract symbol processing and high-level cognition [47], some researchers in AI have later realized that the body plays a crucial part in the emergence of adaptive and intelligent behavior [8, 48–52]. Intelligent behavior is not the result of disembodied computational processes occurring in the brain, but rather that of a complex, dynamic interaction and interdependency among brain, body, and environment [1]: this idea falls under the umbrella terms of embodiment and embodied cognition, conceptual frameworks that have brought fresh ideas to the AI field, putting together a number of different ideas, results, and observations into a unified perspective. In this view, the morphology of an agent, long overlooked, simplified to the bare essential, suppressed in order to be precisely controlled and subjugated by brain, now assumes a role of paramount importance [8, 53].

In the embodied cognition view, not only the body constitutes the mean for a rich and meaningful interaction between the brain and the environment, but also plays an active role in the generation of adaptive and intelligent behavior.

There are many famous examples [8] suggesting how the body can actively contribute to generating effective behavior, in some cases even in presence of no brain or controller whatsoever [54]. But provided that the body plays an important role in intelligence, why should we consider a soft body in particular?

There are a number of cases in point [55] for the fact that a soft body may represent a better substrate for the emergence of intelligent and adaptive behavior.

A first, superficial observation is that the only instances of intelligence we know (i.e. those found in the biological world) largely feature, at least to some extent, soft bodies. Beyond this observation, however, there are deeper insights on the relationship between a soft morphology and intelligence.

First, a soft body can provide a better mean of interaction between brain and environment: while in rigid robotics body dynamics are typically suppressed, the interaction of the robot with the environment limited to the bare minimum, the sensory stimulation scarce, soft robots provide an inherently rich, complex, and dynamic interaction with the environment. By deforming every time they move or interact with the environment, each of these occasions provide an opportunity to gain precious information about themselves and the surrounding environments.

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Moreover, softness comes into play in facilitating the emergence of adaptive and intelligent behavior by alleviating the computational burden of the brain/controller. Put in more engineering terms, there are at least two ways in which morphology can simplify control [8].

A suitable body can take care of low level aspects of the interaction with the outside world, without the need for the brain/controller to take care of these aspects through explicit planning/computation. Control complexity can be outsourced to the body [8].

For example, while grasping an object with a rigid metallic hand requires fine control of grasping force and contact points, a properly designed soft gripper can achieve a stable grasp more easily. A rough high level controller is all it takes in order to close the gripper: then, a stable grasp is facilitated by the friction offered by the deformable material and its ability to passively conform to the object, offering a distributed contact surface rather than isolated contact points. The same holds for other tasks, such as locomotion. The locomotion of a traditional rigid robot on uneven terrain is very complex, requires lot of fine control, and is typically not very robust. On the other hand, many robots characterized by soft limbs can achieve extremely robust locomotion even without a closed loop controller [37, 56, 57].

This idea is sometimes referred to as morphological computation [58, 59], that is: if a brain-less robot (e.g. a robot featuring no sensors and just open-loop control) is solving a task that is thought to require some amount of computation, its morphology must be performing some kind of computation to achieve the same functionality2. This is a rather implicit/indirect notion of morphological computation, but there actually is a more explicit way of interpreting and characterizing the potential contribution of a soft body in terms of computation, showing how morphology can ”compete” on the same ground as the brain.

It has been shown that a soft body can represent a computational reservoir that can be exploited to perform useful computation, thus alleviating the computational burden of the brain/controller [60–62]. In this case explicit computation required to map sensory inputs into motor outputs is outsourced to the soft morphology [8].

This has been shown borrowing ideas from the field of reservoir computing [63], which includes information processing models such as echo state networks [64] and liquid state machines [65]. The main idea of reservoir computing is to exploit the complex

2

In this extreme example morphological computation completely replaces brain computation, allowing a brain-less robot to achieve some useful behavior. The same ideas can be adapted to the more realistic scenario in which the robot indeed embeds some brain computation as well: in that case, morphological computation does not entirely replace brain computation, but can help reducing it.

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dynamics of a given dynamical system (the reservoir, e.g. a set of randomly connected recurrent neurons) to perform computation. Being fed into the reservoir, the inputs can be temporally integrated, combined in a non-linear fashion, and mapped into higher dimensions. The state of the reservoir can then be sampled, and mapped into the desired output state by a simple readout, which can be trained linearly. This thus entails that non-linear and dynamic learning problems can be transformed into linear and static ones.

In [60, 61] it is shown how reservoir computing principles can apply to the case of morphological computation with compliant bodies. The key idea is that if the body of an agent exhibits complex non-linear dynamics, there is no need to implement the reservoir as an abstract collection of randomly connected units: the soft body itself can act as a reservoir. The authors show how soft bodies (represented by generic mass-spring systems) can carry out part of the computation needed to map continuous sensory input into continuous motor outputs, thus potentially alleviating the computational burden of the brain. This framework of morphological computation also points out that in order to boost the computational power of a soft body, the morphology should be ”sufficiently complex”, which goes exactly against the traditional attempts of robotics engineers to simplify robot bodies in order to more easily control them [66], and agrees, instead, with current trends in soft robotics and biorobotics 3_{. It is to be noted that in order to be}

able to exploit this potential for computation, the dynamics of the soft body must be sampled, which requires sensing capabilities that were until very recently outside the reach of soft sensing technologies.

1.4 Evolutionary Computation

Evolutionary Computation (EC) [67, 68] refers to a set of algorithms (Evolutionary Algorithms, or EAs) inspired to different extents to the functioning of natural evolution, namely the combination of natural selection and descent with modification. There are different flavors of EC, but most of them share these characteristics: 1) they are based on populations of candidate solutions to a problem; 2) they use a fitness metric to quantitatively evaluate the performances of each candidate solution; 3) they apply some selection criterion which sees more fit solutions survive and reproduce with higher probability, creating offspring which are randomly perturbed copies of themselves; 4) they are iterative in nature, operating over a number of generations.

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Overall evolutionary algorithms try to capture the algorithmic essence of natural evolution, a trial-and-error process in which innovation is driven by the non-random selection of random variations.

Starting from a (usually) randomly generated initial population, all individuals in the population are evaluated at every generation through the fitness function, usually assigning a scalar number to each individual. Selection is then applied in order to single out with higher probability solutions with higher fitness, discarding those with a lower one. Some of these selected individuals can sometimes survive untouched to the next generation (elitism), while some will be randomly modified (mutation) and/or combined with other selected solutions (recombination) in order to produce new offspring to be evaluated at the next generation. By repeating this iterative procedure long enough, increasingly good solutions start to appear in the population, until the algorithm stops after a termination condition is reached (maximum number of generations, maximum execution time, or thresholds based on maximum or average population fitness). It is to be noted that in light of their biological inspiration, EAs can be used in two different ways: 1) with a more biological flavor, with the aim of modeling and studying some aspects of evolutionary dynamics in silico (especially in fields such as artificial life and computational biology ); 2) as powerful numerical optimization tools, in the context of different engineering problems and domains. EAs have demonstrated to be extremely powerful in both scenarios, and in this work there will be examples of both types of applications.

In addition to a number of hyperparameters to be tuned (namely, those governing selection, mutation and recombination), there are a number of other important aspects to take into account when applying EAs. First, the experimenter needs to select a representation (or encoding) for the candidate solutions, which determines the structure of what is called the genotype, in analogy with the genetic material of living beings. This is usually referred to as the ”blueprint” of the individual (i.e. as the DNA in biology, the genotype contains the instructions to ”build” an individual) and it is usually domain specific: some possible encodings are bit-strings, vectors of real numbers, graphs, etc. Once an encoding is selected, mutation and recombination operators must be designed to match this representation, in that they should be able to manipulate candidate solutions in a meaningful way. While mutation and recombination operators operate on genotypes (loosely mimicking what happens to the genetic material of living beings during reproduction), fitness evaluation and selection operate on phenotypes, each phenotype being the collection of observable features of an individual, determining its success (or failure) as a solution to the problem to be solved. Consequently, another component of an EA is the so called genotype-to-phenotype mapping, i.e. how the genotype gets

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transformed into a phenotype that can be evaluated through a fitness function. This mapping can be direct, i.e. one to one, or indirect. Especially in fields where EAs are applied with a more biological flavor (and in evolutionary robotics as well, Sect. 1.5), this mapping is of paramount importance, and is designed in order to mimic aspects of biological development : this is the reason why indirect encodings are also known in these contexts as developmental encodings, or even as techniques of artificial embryogeny [69]. In this work we will mostly apply Genetic Algorithms [70, 71], which are a popular class of EAs. Applications of simple direct encodings will be reported first, followed by more complex evolutionary systems in which developmental encodings are, instead, adopted.

Figure 1.2: Pioneering approaches to evolutionary robotics. Conceptual setup and physical implementation of one of the first experiments by the Sussex group (a,d) (reprinted from [72]) and the EPFL group (b,e) (reprinted from [73]). c) Brain-body co-evolution in simulated environments (reprinted from [74]). f) Bringing robots whose brains and bodies were evolved in simulation into the real world (reprinted from [24]).

1.5 Evolutionary Robotics

Evolutionary Robotics [12–15, 75] (also known as ER, or evo-robo) is an interdisciplinary research field borrowing ideas from and contributing to a number of different disciplines, among which robotics, artificial life, cognitive sciences, and evolutionary biology. At the core of evolutionary robotics is the application of evolutionary algorithms in order to automatically design intelligent and adaptive robots.

The idea was put forward in 1984 already by the neurophysiologist Valentino Braitenberg [76] in the form of a thought experiment, and partly investigated in simulation in the

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early nineties already [77]. In 1994 two independent research teams —one surrounding Floreano and Mondada at EPFL, the Swiss Federal Institute of Technology in Lausanne, another involving Cliff, Harvey, and Husbands at University of Sussex in Brighton— reported the first successful cases [72, 73] of artificially evolving neural circuits for robots with fixed morphology operating in real environments. In the same year, Karl Sims was investigating the co-evolution of brains and bodies in simulated environments [74], but the field had to wait until year 2000 before Lipson and Pollack [24] were able to bring the results of brain-body co-evolution into the real world, thanks to 3D printing (Fig. 1.2). The main ingredients of an evolutionary robotics experiment are the following:

• An environment (either the real world or a physically plausible simulated one) and a task that we want the robot to solve (e.g. walk as far as possible). The combination of these two is often referred to as task environment.

• A population of robots, some aspects of which should be optimized to meet a certain goal (i.e. they will be under evolutionary control ).

• A fitness function, measuring how well each robot performs (e.g. speed)

• An evolutionary algorithm (Sect. 1.4), manipulating the robot population over a number of generations.

A peculiar aspect of evolutionary robotics (as opposed, for example, to techniques for robot learning), is that in its most general form ER tries to automatically design complete robots, dictating not only the controller but also the morphology, sensory

system, etc. This is particularly interesting for a number of reasons.

First, as postulated by the embodied cognition theory (Sect. 1.3), intelligence is not centralized in the controller, but rather emerges from a complex intertwining of different morphological, neurological, and environmental factors: the possibility to co-optimize all aspects of a robot brain and body is thus particularly appealing, in order to discover effective combinations of the aforementioned factors for a given task environment. In many cases it is not trivial to imagine which morphology, sensor placement, and controller will best solve a set of tasks while satisfying a number of engineering requirements. The ability of evolutionary algorithms to ”think outside the box” is well known in the EC and ER fields ([8] reports several examples). Desirable in most cases, this form of ”artificial creativity” sometimes gives rise to unwanted phenomena such as the one commonly referred to as perverse instantiation, in which artificial evolution produces solutions that indeed maximize the fitness function, but do so in a completely non-intuitive and often

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undesirable way4. This kind of phenomenon, although unwanted, highlights interesting properties of these algorithms which, similarly to natural evolution, are able to adapt and exploit all opportunities offered by a given ecological niche. As we will see in what follows, in light of this ability of evolutionary algorithms to ”find shortcuts”, they are able to systematically discover and exploit simplification mechanisms such as embodied intelligence and morphological computation automatically and in a variety of tasks environments, as opposed to the heuristic procedures and hand-crafted solutions often adopted by human designers. If it is possible to solve a task in a simple way (e.g. devise a suitable morphology that can perform self-stabilized walking in presence of a simple activation), artificial evolution will usually try to do that first, before evolving more complex strategies (e.g. generating a complex actuation strategy for a robot with not-suitable morphology).

Second, as we have mentioned earlier (Sect. 1.1), with soft robotics more dimensions are being added to the design space, each one potentially contributing in a non-intuitive way to the success or failure of a robot for a particular task: the availability of a completely automated design automation technique able to explore this increasingly large design space and find good trade-offs between multiple design requirements and constraints is thus extremely desirable. Third, from an artificial intelligence and artificial life perspectives, it is appealing to fix a priori as few aspect of an agent as possible, in order to avoid introducing human biases into the design [8] and to observe how artificial creatures will evolve and develop autonomously under different conditions.

1.6 Evolution in simulation and in the real world

Researchers in embodied cognition stress the importance of testing hypotheses related to the emergence of intelligent behavior in the real world rather than in simulated environments [48, 50]. As elegantly stated by Rodney Brooks in [78], ”the world is its own best model”.

An artificial evolutionary process, however, requires testing a very large number of designs, which is very impractical to do in the real world: it is time consuming, requires very

4

Evolutionary algorithms are excellent at finding bugs in programs and physics simulations. If there is a bug which can be exploited in order to artificially maximize fitness, evolution will often find it. For this reason, it often takes a number of preparatory evolutionary runs before we are able to perform a meaningful evolutionary experiment, the first ones deemed to fail due to evolution exploiting undesirable states of the simulation. Apart from domain specific ones, some common ”glitches” exploited by artificial evolution in physics simulations are simulation instabilities (e.g. high velocities or impulsive forces are generated that cannot be correctly integrated with a given time step, resulting in unrealistically long jumps, exploding body parts, and other artifacts that can lead to huge fitness scores in a non legitimate way), errors in the friction model (e.g. which can result in robots that manage to move about or glide onto the ground in a non realistic way), errors in the collision models (e.g. resulting in robots that manage to move across obstacles or through the ground plane).

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reliable hardware able to withstand wear and tear (usually not the case in soft robotics, where technologies are in most cases not mature yet), as well as the design of additional experimental apparatuses.

Despite these limitations, it is indeed possible to set up an evolutionary experiment where the evaluation of each individual occurs in the real world.

However, while this can suffice in order to produce effective enough behaviors [79–82], evolution in the real world usually entails important limitations to the number of fitness evaluations that can be done (i.e. the population size and the number of generations), so that studies that aim at testing artificial life or artificial intelligence hypotheses often become outside the reach5. Most importantly, with current technology, evolution in the real world greatly reduces the number of aspects of a robot that can be put under evolutionary control, often to the sole controller, which can be limiting from an artificial life/intelligence point of view in light of the important role of morphology (Sect. 1.2).

Traditionally, the number of morphological traits that can be evolved in the real world is usually very limited due to technological constraints. A possible way to overcome this limitation is to resort to modular robotics [83]. As for soft robotics, although progresses in 3D printing and soft fabrication [19, 20] will soon allow to evolve and test complete soft robots, the aforementioned limitations in terms of number of fitness evaluations will still hold.

The solution probably lies in hybrid approaches [22–24] which combine simulation and evaluation in the real world, while mitigating at the same time the so called reality gap problem [84], i.e. the risk that behaviors evolved in simulation will not maintain their efficacy once implemented in the real world due to discrepancies between the real and the simulated environments.

For the role that morphological evolution will have in the continuation of this work, and in light of our intended purposes, we will accept simulated worlds as good enough surrogate models of the real one, and will not touch upon techniques for evolving robots directly in the real world or for overcoming the reality gap.

The studies reported in what follows have thus been conducted by evaluating robots in simulated environments6_{, using various modeling techniques and tools which range}

5

In order to be able to do hypotheses testing through evolutionary simulations several repetitions are required for each treatment in order to account for the stochasticity of these algorithms. Also, it is always desirable to have large populations, and to let them evolve for several of generations. All of this quickly becomes unfeasible when fitness evaluation is performed in the real world.

6

There will be, however, one example of successfully transferring a behavior from simulation to the real world.

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from custom dynamics models to more standard robotics physics engines allowing the simulation of soft materials [85] or compliant elements [86]7.

1.7 Evolutionary developmental soft robotics

(evo-devo-soro)

As the term implies, evolutionary developmental soft robotics (evo-devo-soro) refers to the application of computational techniques inspired by natural evolution and development in order to automatically design soft robots (Fig. 1.3).

In addition to the already mentioned evolutionary approaches, in fact, some of the last experiments reported in this thesis also introduce developmental paradigms for soft bodied creatures, following an approach (evo-devo [90]) in which evolution not only dictates the innate traits of evolving creatures, but also the parameters of developmental processes that modify some aspect of the robot morphology during their lifetime. Before delving into the core and original aspects of this work, a few other approaches to evo-soro or evo-devo-soro are worth mentioning.

Considerable contributions to this research niche come from Rieffel et al. [91–95]. The authors have experimented with different flavors of evo-devo-soro, including evolving gaits or material properties for soft robots (even real ones) with a fixed morphology (spiking neural networks were used for gait control [96]), evolving free-form soft morphologies (represented as tetrahedral meshes), as well as introducing grammatical developmental encodings capable of operating on such mesh-based representations. In a different context, they have also dealt with the evolution of a particular class of soft robots, tensegrity structures [97] — both in simulation and in the real world [98], and explored aspects related to morphological computation [99] and morphological communication [100]. Hod Lipson’s research group at Cornell University has greatly contributed to advancing this field. From this group, Hiller et al. have approached the problem of evolving continuous, amorphous, multi-material morphologies [101], that were volumetrically actuated. In this context they have experimented with various indirect encodings, such as the Discrete Cosine Transform (DCT) representation, Gaussian Mixtures [107], and Compositional Pattern Producing Networks (CPPN) [108]. Additionally, they have applied evolutionary techniques to the design of soft robots and mechanisms that were later fabricated and validated in the real world [21]. Lastly, they have also developed effective simulation tools for designing and analyzing soft robots (VoxCAD [85]) that

7

For the sake of completeness, some other simulation frameworks for soft robots — not used in this work — are the following: [87–89]

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Figure 1.3: Some results in the field of evo-soro / evo-devo-soro. a) An evolved multimaterial soft robot and its physically fabricated version (b), locomoting inside a pressure chamber (right). Reprinted from [21]. c) Two multi-material soft robot evolved in a simulated environment, reprinted from [101]. d) Another example of a multimaterial soft robot evolved in a similar setting, reprinted from [102]. e) An evolved robot composed by rigid components (white) interconnected by muscle-like linear actuators (red strings). Reprinted from [103]. f) A soft robots composed of rigid components (white) interconnected by muscle-like soft actuators (red). Reprinted from [104]. g) An example of an evolved multicellular soft animat, reprinted from [105]. h) Another example of a fine-grained multicellular creature from the morphogenetic engineering

field, reprinted from [106].

will come into play later on in this work. Building on these tools and techniques, Cheney et al. (from the same research group at Cornell University), have extended the scope of these results by further elaborating on the use of CPPNs as a general indirect encoding for soft robots [102], by evolving soft robots for different tasks (such as squeezing into apertures [40]) and introducing different actuation mechanisms [109]. Recently they have focused on broader aspects related to the difficulty of co-evolving brains and bodies [110]. From a different research group, Methenitis et al. applied similar tools to evolve soft robots in altered gravity conditions [111], having space exploration as a potential target application8.

Lessin et al. have explored the evolution of virtual creatures composed of rigid body segments and muscle-like actuation [103] (also done by [114]), a concept later extended in [104].

Contributions to the evolution and development of soft-bodied creatures also come from neighboring research fields such as morphogenetic engineering [115]. By using considerably more complex and fine-grained models, Joachimczak et al. have widely explored the

8_{Interestingly, [111] and [112] simultaneously introduced the use of an open-ended evolutionary}

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brain-body co-evolution and development of soft-bodied multicellular animats using 2D physics models and gene regulatory networks [105, 116–118]. Doursat et al. [106] report contributions of a similar flavor, where walking behavior was achieved in a 3D simulated environment with fine-grained multicellular soft robots.

1.8 Structure of this thesis

This thesis organizes research results from a number of journal and conference articles published over the course of three years. These were aggregated and structured into five main chapters, which deal with different aspects of soft robotics, embodied intelligence, and, of course, evolutionary and developmental robotics.

The complexity — both technical and conceptual — of the experiments and evolutionary systems presented in each chapter gradually increases, from Ch. 2, were a simple genetic algorithm with direct encoding is used as an optimization tool for engineering purposes, to Ch. 6, where a powerful and comprehensive evo-devo system is introduced, allowing broader and more general investigations. A similar progression is followed by soft robot models and simulators used in each chapter, moving from custom 2D models to full-fledged 3D simulators.

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Parameters estimation and

locomotion of an aquatic soft

robot

1

Abstract

In this chapter a biomimetic aquatic soft robot will be introduced, that provided a real case study and application scenario for some of our investigations on evolutionary soft robotics and embodied intelligence. The robot has attracted our attention for its complex soft morphology, its rich interaction with the underwater environment, as well as for its ability to exhibit complex and effective behaviors despite the absence of a brain or controller whatsoever: in many regards, it represents a perfect case study for investigating some aspects of embodied intelligence. After describing the robotic platform and its associated model (that will come into play in the next chapter as well), we will show how evolutionary algorithms were applied in order to achieve a faithful mathematical model of its locomotion dynamics through a parameters estimation procedure, where more conventional identification techniques could not be applied. The model was then used to:

1_{Appeared as:}

• Giorgio-Serchi, F., Arienti, A., Corucci, F., Giorelli, M., & Laschi, C. (2017). Hybrid parameter identification of a multi-modal underwater soft robot. Bioinspiration & Biomimetics, 12(2), 025007.

• Calisti, M., Corucci, F., Arienti, A., & Laschi, C. (2015). Dynamics of underwater legged loco-motion: modeling and experiments on an octopus-inspired robot. Bioinspiration & Biomimetics, 10(4), 046012.

• Calisti, M., Corucci, F., Arienti, A., & Laschi, C. (2014). Bipedal walking of an octopus-inspired robot. In Biomimetic and Biohybrid Systems: Third International Conference, Living Machines, Proceedings (pp. 35-46). Springer International Publishing.

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1) highlight properties related to the embodied intelligence of this robot, arising from the rich and dynamical interaction between the soft body and the aquatic environment; 2) identify faster morphological arrangements, one of which demonstrated to transfer to the real world, entailing a considerable improvement in the locomotion speed of the robot; 3) enable general studies on underwater legged locomotion.

2.1 Introduction

The PoseiDRONE robot [37] (Fig. 2.1) is an octopus-inspired aquatic soft robot, that was developed by combining design principles from soft robotics, embodied intelligence, and biomimetics. A careful design and a clever exploitation of the intrinsic compliance of the platform are key to its ability to exhibit a robust and effective behavior despite its brainless nature.

Figure 2.1: The aquatic soft robot PoseiDRONE [37] while crawling (a) and swimming (b). Credit to ”The Age of Robots” by M. Brega.

The robot combines two units: a pulsed-jet thruster for waterborne swimming (Fig. 2.2), and a legged module for crawling onto the seabed (Fig. 2.3). Both subsystems, made to a large extent of soft silicone, are mounted onto a rigid, cross-shaped aluminum frame. The overall weight of the robot is 0.75 kg, and 80% of its volume is composed of silicone, providing the robot with a remarkable structural flexibility.

The swimming unit (Fig. 2.2) consists of a soft, hollow, silicone mantle, which periodically contracts and expands due to the action of a number of radially-arranged internal cables, which are actuated by a single rotational motor receiving a constant voltage. The mantle injects ambient fluid while expanding, and ejects it when contracting, which results in a net thrust. Interestingly, while contraction is determined by active cable traction, the

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Figure 2.2: Schematic representation of the PoseiDRONE swimming unit (lateral view). Highlighted are: (1) cable attachment points over the elastic shell, (2) axial location of the cross sections subject to cable traction, (3) the nozzle, (4) inflow valves, (5) the motor, (6) the crank, (7) the axial pulley which distributes the cable over the

various cross sections, (8) the cables.

subsequent expansion is achieved thanks to the passive elastic response of the mantle, which naturally returns to its original undeformed position.

Figure 2.3: Schematic representation of the PoseiDRONE crawling unit (lateral view, only two of the four limbs are shown). Highlighted are: (1) the flexible limbs, (2) interconnection between the steel cable and the silicone limb, (3) the steel cable, (4) spherical bearing (allows translation and rotation), (5) the motor, (6) the crank, (7) the motor case and (8) the aluminum frame supporting the limbs and the thruster. The figure also highlights the center of mass (CoM) and the center of buoyancy (CoB) of

the robot, which are decoupled.

The crawling unit (Fig. 2.3) is composed by four conic silicone limbs, each one actuated by a dedicated rotational motor (receiving a constant voltage) through a three-bar mechanism (Fig. 2.4). The soft limbs gently absorb asperities and impacts with the ground, allowing the robot to walk on unstructured grounds in absence of closed-loop control. Additionally, the fact that the center of mass (CoM) and center of buoyancy (CoB) are decoupled (as shown in Fig. 2.3) produces a rotational momentum which tends to passively bring the robot back to an equilibrium position after every contact with the ground.

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Figure 2.4: A schematic of the three-bar mechanism during the actuation routine: (a) side view and (b) top view. The elements depicted represent: (1) the flexible limb, (2) distal part of the steel cable and interconnection with the soft limb, (3) the steel cable, (4) the spherical bearing for the steel cable, (5) the motor, (6) the crank, (7) the motor case, (8) the aluminum frame supporting the limbs and the thruster, (9) circular loop controlled by the motor (10) eccentric loop achieved at the distal part of the limb

through the three-bars mechanism, suitable for pushing onto the ground.

The observations above provide good evidences of how a design based on embodied intelligence principles can result in a remarkable simplification of the control (Sect. 1.3), which is in this cased reduced to a constant open-loop activation. All the complexity required to perform robust locomotion in unstructured environments is outsourced to the body thanks to a clever mechanical design.

2.2 Methods

2.2.1 Modeling

The first step towards the engineering characterization of a robotic platform entails mathematical modeling, which was dealt with separately for the two modes of locomotion, as briefly described below. This section also serves as an example of one of the many possible approaches to modeling soft robots, which differs from others that will come into play in the following chapters. Here, instead of trying to precisely capture all aspects of the complex dynamics arising from the interaction of a soft body with the environment, a more synthetic approach is adopted: in the spirit of simplification, fine details are neglected and effects combined when possible, trying to achieve a simple, interpretable, and computationally-efficient model.

2.2.1.1 Swimming dynamics

The swimming dynamics of the robot are rather simple, and can be described by equations governing the motion of a neutrally buoyant body swimming along the surge direction x (Eq. 2.1):

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˜

m¨x = −Λ ˙x| ˙x| + τ (2.1)

where ˙x is the velocity in the surge direction and ˜m = M + mas is the effective mass

comprising of the inertia of the robot M and its added mass mas during swimming.

The drag acting on the swimming vehicle is based on the ensemble viscous quadratic drag coefficient Λ, while τ is the pulsed-jet thrust generated by the cyclic routine of inflation and deflation of the silicone shell. Modeling τ is the real modeling challenge here: however, as we will focus on crawling dynamics, this aspect will not come into play in what follows, and will not be detailed. The interested reader can refer to [119].

2.2.1.2 Crawling dynamics

It is worth detailing the model of robot’s crawling dynamics, as we are going to utilize it for some of our investigations. The locomotion of the robot is mainly planar i.e. the CoM approximately moves along the xy plane (Fig. 2.6). A simplified sagittal model was thus developed, instead of one that accounts for all the degrees of freedom of the real robot. The model comprises a central body with three DoF (two translations, one rotation) and four legs, all of which are immersed in water, Fig. 2.3. Each compliant leg is dynamically modeled as a massless spring-damper system, whose kinematics is derived from the three-bar mechanism described in [120]. Dynamics equations are detailed below:

˜ mcx¨ = " ₄ X n=0 tn(Fkx+ Fcx)n+ tn(Ftx+ Fnx)n # + Fdrx (2.2) ˜ mcy¨ = " ₄ X n=0 tn(Fky+ Fcy)n+ tn(Fty + Fny)n # + Fdry+ Fg+ Fb (2.3) J ¨ϑ0 = " ₄ X n=0 tn(Mk+ Mc)n+ tn(Ms)n # + Mb+ Mdr (2.4)

Eq. (2.2, 2.3) account for the translations experienced by the robot in the sagittal (xy) plane. On the left hand side (LHS) of these equations, ˜mc= M + mac represents the

effective mass during crawling, which differs from the swimming effective mass due to the crawling added mass term mac.

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The right hand side (RHS) terms in square brackets in Eq.2.2, 2.3 account for reaction forces arising from ground contacts (Fk, Fc) and fluid resistance, referred to here as

sculling forces, (Ft, Fn).

The reaction forces have an elastic (Fk) and a damping (Fc) component. For example, in

the x direction, Fkx = k dx, while Fcx = c d ˙x, with dx representing the compression

state of the limb in x. The terms k and c = 2 dr√kM respectively represent the elastic and damping coefficients. Additional parameters related to the interaction of the limbs with the ground are the static (µs) and dynamic (µd) friction coefficients, which do not

appear in these equations (see [120] for additional details).

The viscous drag experienced by each of the four limbs is divided in a tangential (Ft)

and normal (Fn) component (relative to the radial axis of the arm), which depend,

respectively, on the associated tangential (λt) and normal (λn) drag coefficients, as in

[121]. The viscous term of each individual limb (sculling forces) is summed to the force Fdr, which accounts for the drag experienced by the central body as it moves along the

surge and heave directions. For example, in the x direction, Fdrx =

Λt

2 x| ˙˙ x|.

The tn coefficients, updated by a routine detecting collisions between each limb and the

ground, determine which forces act on the body at a given instant in time. When a limb is in contact with the ground (tn= 1, tn= 0) reaction forces are included and sculling

forces are neglected. When a limb is detached from the ground (tn= 0, tn= 1), contact

forces are not computed, with sculling forces being accounted for instead.

Dynamics in the vertical direction y, Eq. 2.3, include body forces such as gravity (Fg)

and buoyancy (Fb). These depend on the mean density of the robot (ρr) which in turn

accounts for both material properties and extent of inflation of the buoyancy module. The incorporation of the separation distance b between the CoM and the CoB and their relative orientation β with respect to the median plane (Fig. 2.3) enables to account for dynamics that are peculiar to underwater legged locomotion [120, 122].

Finally, Eq. 2.4 describes the torques about the axis z orthogonal to the sagittal plane, determining the pitch of the robot (ϑ0). The quantity J which appears in the LHS of

the equation represents the aggregate inertia of the body. Relevant moments appearing in the RHS are Mk, Mc, Ms, Mb which respectively represent the moments arising

from elastic, damping, sculling and buoyancy forces. An additional term Mdr which

accounts for viscous forces acting against body rotations about the z axis is also taken in consideration.