Modeling embodied intelligence: can we capture its essence by modeling internal and external interactions?

In soft robotics, embodied intelligence leverages on the compliance of the physical body to let sensory-motor behaviour emerge from the interaction with the environment. Soft robot modeling often focuses on the effect of actuation on the deformable body. Can we model and describe mathematically the deformations of a soft robot body under external interaction forces? Would that capture the essence of embodied intelligence? We argue that addressing this challenge, in an interdisciplinary effort, would make embodied intelligence and soft robotics take a leap forward and transition from a trial-and-error pioneering phase to a model-informed discipline era.


Introduction
Embodied intelligence is the emergence of sensory-motor behaviour from the complex interaction of the body with the environment [1].In soft robotics, such external interactions are received by the compliant body as deformations, which are combined with the deformations generated by actuators.
Modeling these internal and external interactions into a unified representation would allow to grasp embodied intelligence, for design and control purposes.A mathematical description of the internal and external interaction forces is crucial for soft robots to achieve controllable and programmable performance [2].Indeed, we are at a stage where the field of soft robotics, after an impressive growth [3], faces the challenge of evolving into a well-rooted and principled scientific discipline that provides tangible societal benefits, thereby thriving over the next few decades [4].At the same time, computational modeling has progressed towards techniques and tools that make complex models and simulations affordable.
On the other hand, keeping pursuing prototype-based design and control methodologies, without a supporting engineering modeling framework would inevitably prevent soft robotics from becoming the expected thriving scientific field with clear societal benefits.Prototype-driven design (also referred to as trial-and-error) was likely the fastest and most efficient way to proceed, especially in the exploratory phase.Today, we are at a stage where computational modeling within a model-driven umbrella can be achieved and can enable (i) scaling up soft-robot design and control in response to application needs, and (ii) replicating the behaviour of a soft robot by leveraging simulation and datadriven modeling, opening the path towards soft-robot digital twins.

Bibliometric ground
Figure 1 shows the percentage of computational modeling adopted from 1950 to 2020 in the aerospace sector, where it is prominent, in soft robotics, where it is emerging, and in the entire research community, including the humanities and other sectors where computational modeling is almost non-existent.We note how computational modeling kept growing in the aerospace field, bringing a series of tangible breakthroughs that include the Apollo program and space flight, along with the exponential growth of civil aviation, and possibly space tourism.We believe that soft robotics might be at a stage similar to the "Apollo years", where the community must adopt a model-driven workflow for design and control, moving forward from prototype-driven and often trial-and-error methodologies.We argue that this evolution can be achieved by a more pervasive use of rigorous computational mathematical modeling to describe internal and external interactions, and their coupling.This can lead to a flexible hierarchy of improved models, from high-to low-dimensional (or fundamental models) that can thoroughly capture embodied intelligence and can be used for design and control purposes, thereby reducing the sim-to-real gap.
Figure 1.Growth of modeling in aerospace (blue), soft robotics (red) and the whole research community (black) in terms of number of publications with computational modeling, normalized by number of publications per sector; the y axis is the percentage of publications including computational modeling with respect to a given sector.Dashed lines correspond to actual publications with computational modeling per year, while the solid lines are ten years moving averages.Ten years batches are commonly considered research cycles and represent a reasonable approximation to estimate a trend in a given sector.We note that the common exponential growth shadowing the trend of every field in the past few years is not present thanks to the normalization adopted, that emphasize the percentage of computational modeling in a given discipline.The search for publications was done in Scopus, and the search details are as follows for each field: "All literature":

Computational modeling for soft robotics
Computational modeling involves using computer simulations of physical principles translated into mathematical models, to provide accurate, efficient, and flexible proxy solutions of the underlying real-world.
In other fields, mature computational modeling has enabled engineering tasks that were previously unimaginable -the aerospace and biomedical fields (e.g., cardiovascular dynamics modeling) constitutes two notable examples [5].The computational modeling challenge in soft robotics is considerable, as the range of morphologies (e.g., arms, fingers, legs, and fins), physics (e.g., materials, fluids, etc.), scales (from few mm to few m), abilities (e.g., reaching, grasping, walking, morphing, growing, swimming, jumping, crawling, and digging) and intended applications (e.g., healthcare, manufacturing, underwater sensing and manipulation, scientific exploration, entertainment, and more) is extremely diverse.
Computational modeling in the context of soft robotics inherently leads to multiscale and multiphysics problems, given the diverse configurations that can be adopted.Multiscale and multiphysics are notoriously challenging but have successfully been achieved in some fields, such as aerospace engineering.It is challenging due to a number of reasons: i) multiscale dynamics in space and time often necessitates high-dimensional computational representations, which quickly become intractable (e.g., in turbulence, resolving all relevant scales requires supercomputer simulations); ii) nonlinearity inherent in the dynamics, for example convective nonlinearity in fluid dynamics; iii) partially known interfacial physics and unmodeled dynamics (nonlinear friction, turbulent drag, etc.); iv) ill-conditioned systems of equations.Therefore, it is not surprising that interdisciplinary efforts are required.
Modeling and computational techniques that are used in other sectors can be applied in soft robots, with mutual benefit for soft robotics progress and research opportunities for a range of communities, including applied mathematics, computer science, and computational physics, to name a few.Specifically, soft robotics progress would benefit from further research efforts in 1) modeling the internal interactions of soft robot components that lead to efficient movements and deformation; 2) modeling the robot interactions with the surrounding environment; and 3) modeling their coupling with internal interactions, or embodied intelligence.These computational models could then be used for instance in the context of automatic robot design and optimization [6], where material, geometrical configuration, and actuation/sensing, can all constitute the parameter space that one optimizes to efficiently achieve the desired tasks.
While the first point, i.e., modeling the internal interactions of soft robots, has been addressed in soft robotics, especially for control purposes, modeling external interaction is under-investigated.We main envisage three broad families of approaches: (i) based on continuum mechanics; (ii) based on lumped parameters; (iii) data-driven.They can all be considered both for the case of interaction with fluids (relevant for swimming, flying, and more) and solids (relevant for locomotion and manipulation).Despite the differences among these approaches, they can be formalized into one general equation, as proposed in [7] and reported here to the reader's convenience, whose terms vary with the approach taken.
=   +   +    Ω  where  is a differential operator,   describes the soft body () in the soft body domain Ω  ,   is a non-linear term that describes the soft-body mechanics,   and   are coupling terms that accounts for internal and external interactions, respectively.This unified approach may allow significant improvements in soft robot performance for tasks including grasping, walking, crawling, underwater and underground operations, and morphological adaptation, among others.It might also open novel and sought-after robot abilities and applications, such as next-generation industry and services and human-robot interaction.

Conclusions
We highlighted the need and the challenges in modeling internal and external interactions in soft robotics.All this in the quest for mathematically describing the complex interaction of a soft robot with its environment and capturing the underlying principles of emergent behaviour in embodied intelligence.Is such an effort worth?It is, to the extent that it enables model-informed design and control of soft robots.Whether a computationally obtained design could be physically realized should be considered under two perspectives: on one side, ready-to-build soft robots could be designed with available sensors and actuators only.On the other hand, a particularly effective design might steer the development of novel actuation and sensing technologies.
Soft robots that can fulfil given design inputs, fully leveraging on embodied intelligence, can deliver unprecedented abilities, and can respond to unmet needs.Furthermore, computational models can be used to construct digital twins, for real-time monitoring and control of the soft robot asset during a mission or throughout its life cycle.
Soft robotics is at a critical stage of development, and it is the right time for this field to evolve towards becoming a discipline, with a computational modeling framework and structured design methodologies.
Figure1.Growth of modeling in aerospace (blue), soft robotics (red) and the whole research community (black) in terms of number of publications with computational modeling, normalized by number of publications per sector; the y axis is the percentage of publications including computational modeling with respect to a given sector.Dashed lines correspond to actual publications with computational modeling per year, while the solid lines are ten years moving averages.Ten years batches are commonly considered research cycles and represent a reasonable approximation to estimate a trend in a given sector.We note that the common exponential growth shadowing the trend of every field in the past few years is not present thanks to the normalization adopted, that emphasize the percentage of computational modeling in a given discipline.The search for publications was done in Scopus, and the search details are as follows for each field: "All literature": (TITLE-ABS-KEY ((fig* OR tab* OR par* OR an* OR the* OR for* OR wor* OR pape*); "Aerospace": (TITLE-ABS-KEY ((ALL) AND (aeronautic* OR aerospace* OR aerodyn*))); "Soft robotics": (TITLE-ABS-KEY ((ALL) AND ("soft robot*" OR "soft bodi*" OR "soft body"))); "Aerospace Computational Modeling" (TITLE-ABS-KEY((ALL) AND (AEROSPACE) AND model* AND (computation* OR digital*))); "Soft robotics Computational Modeling": (TITLE-ABS-KEY((ALL) AND (SOFT ROBOTICS) AND model* AND (computation* OR digital*))).
-year moving average Soft robotics Soft robotics: 10-year moving average All literature All literature: 10-year moving average