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Soft robotics is an emerging field of research where the robot body is composed of flexible and soft materials. It allows the body to bend, twist, and deform to move or to adapt its shape to the environment for grasping, all of which are difficult for traditional hard robots with rigid bodies. However, the theoretical basis and design principles for soft robotics are not well-founded despite their recognized importance. For example, the control of soft robots is outsourced to morphological attributes and natural processes; thus, the coupled relations between a robot and its environment are particularly crucial. In this paper, we propose a mathematical foundation for soft robotics based on category theory, a branch of abstract mathematics where any notions can be described by objects and arrows. It allows for a rigorous description of the inherent characteristics of soft robots and their relation to the environment as well as the differences compared to conventional hard robots. We present a notion called the category of mobility to well describe the subject matter. The theory has been applied to a model system and analysis to highlight the adaptation behavior observed in universal grippers, which are a typical example of soft robotics. The aim of the present study is not to offer concrete engineering solutions to existing robotics but to provide clear mathematical description of soft robots by category theory and to imply its potential abilities by a simple soft gripper demonstration. This paper paves the way to developing a theoretical background and design principles for soft robotics.

Soft robotics is being intensively studied to overcome the difficulties of traditional robots made from a rigid underlying structure and to create new value by exploiting the intrinsically soft and/or extensible material [

Soft robotics is inspired by biology, such as the muscle–tendon complex, skin sensors, and retina observed in living organisms [

However, the theoretical background of soft robotics is limited to a few studies in the literature, which are introduced below. This is in contrast to hard robots, where a vast and deep theoretical foundation is provided, ranging from conventional and modern control theories [

The challenges for developing a theoretical basis for soft robotics stem from a variety of unique attributes which are difficult to describe within conventional frameworks. First, soft robots have a huge number of degrees of freedom. For example, the number of elements involved in a deformable body can be on the scale of Avogadro’s number. From an energetic point of view and with respect to Fourier space, the degrees of freedom will be much smaller than Avogadro’s number, but still sufficiently large if the precision needs to be at a very fine scale. Second, soft robots need to adapt to the environment and its uncertainty. Environmental information, such as the shape of the object of interest to be grasped, is difficult to acquire precisely prior to physical interactions unless high-precision measurement systems are assumed. Accommodating a diverse range of environments remains to be addressed.

In this paper, we propose an approach to soft robotics based on category theory, which is a branch of mathematics that simplifies all mathematical notions into objects and arrows [

To facilitate intuitive understanding, we discuss a grasping problem as an example application of the theory. This is schematically shown in Figure

Hard and soft robots. (a) Hard robots require precision control of rigid joints to grasp an object, whereas (b) a soft robot autonomously adapts its shape to the object.

The remainder of this paper is organized as follows. In Section

We introduce fundamental concepts in category theory such as

A category is a network formed from composable arrows that intertwine with objects. The objects can be considered to represent some phenomena, while the arrows show a transformation or process between these phenomena. A category is a system consisting of objects and arrows that satisfies the following four conditions:

Each arrow

or

The direction of the arrow does not need to be limited to from left to right; if convenient, it can be from bottom to top, from right to left, etc. A subsystem of the category built up with these arrows and objects is called a diagram.

Assume that there are two arrows

Then, there is a unique arrow called the composition of

Assume the associative law for the following diagram:

Then, we can assume the following:

When any compositions of arrows with the same codomain and domain are equal, the diagram is called commutative.

The last condition is the unit law. For any object

In other words,_{X} =_{Y}

Because of the natural correspondence between objects and their identities, we may identify the objects as identities. In other words, we may consider the objects as just the special morphisms. In the rest of this paper, we may adopt this viewpoint without notice.

A category is a system composed of two kinds of entities called objects and arrows that are interrelated through the notions of the domain and codomain and is equipped with a composition and identity to satisfy associative law and unit law.

There are so many examples of categories in mathematics; almost everything in mathematics can be formulated in terms of categories. For example, the category

Once a category given, we can define the sameness between different objects in terms of isomorphism.

An arrow

Two different objects are considered essentially the same in the category if they are isomorphic. Since they are connected by an isomorphism, if one is in a certain diagram, the other is in a completely similar diagram. This is why we can count things with both fingers and stones; a set of five fingers and set of five stones are isomorphic in the category of sets and functions. The notion of “five” is obtained by recognizing such isomorphisms.

In summary, this means that, every time one category is postulated, a suitable sameness called isomorphism is determined for objects that are contacted by reversible arrows in the category. Here, we present the basic structure for elucidating the sameness between obviously different phenomena.

In this world, it is impossible for any phenomenon to be completely identical with any other phenomenon. Therefore, there is no repeatable phenomenon in an absolute sense. This is the undeniable nature of things. It is only possible for us to discuss a law in the form of saying “the same thing can be said about similar phenomena” when we set up an equivalence relation between different phenomena.

Although each phenomenon is restlessly changing, and hence there are no completely identical phenomena as mentioned above, first we regard given phenomena being approximately constant with respect to some equivalence relation. Furthermore, while each of the patterns of the relationship between the system and environment is physically fundamentally unique and unrepeatable all time, we consider them under some criteria of equivalence.

The concepts of the system and interface between the system and environment, under certain equivalence relations, should be essential ingredients in every science. In physics, these concepts are formulated in terms of algebras of quantities (i.e., observables), and states are the expectation functionals of the algebras. The pair of the algebra of quantities and its state provide a general framework to describe the statistical law for a certain system in a certain relationship with the environment. The state connects the quantities themselves and their values in a spectrum and reflects the condition based on the relationship between the system and environment. The state connects the past and future; it can be defined through the equivalence between preparation processes and provides a basis for discussing the possible transition of the relationship between the system and environment.

The concept of the category of mobility is based on the idea that the state at a moment is an effect caused by hidden dynamics while at the same time causing the future development of the dynamics. In general, such a development cannot be expected to be deterministic. Then, we can consider all possible transitions between states, starting from the given initial state

Let

The notion of the category of mobility provides a theoretical framework for soft robots. It can be used to define hard, soft, and effectively soft robots together with the basic notions of category theory, which are introduced in the next subsection.

A functor is defined as a structure-preserving correspondence of two categories.

A correspondence

It maps

For each

In short, a functor is a correspondence that preserves diagrams or, equivalently, the categorical structure. A functor is a very universal concept. All processes expressed by words such as recognition, representation, construction, modeling, and theorization can be considered to be the creation of functors. Once the notion of a functor is established, it is natural to introduce the concepts of the category of categories and isomorphism of categories.

The category of categories, which is denoted by

The notion of the isomorphism of categories seems to be quite natural, but the concept of “essentially the same” categories is known to be too narrow for formulations in mathematics. To define such essential sameness in mathematical terms, which is called the equivalence of categories, we need the central notion of category theory: natural transformation.

Let

For any

For the natural transformation, we use the notation

It is easy to see that the functors from

The functor category from

Since any category can be considered as a subsystem of some functor category in some sense, all kinds of sameness that we can define are actually formulated in terms of natural transformations, especially natural equivalence.

For natural equivalence, which is the isomorphism between functors, we introduce the notion of the equivalence of categories. This is the functor that represents the essential sameness between categories.

A functor

One remark here is that isomorphism and categorical equivalence are categorical counterparts of homeomorphism and homotopy equivalence.

To define and analyze the notion of soft robots, we begin with the fundamental notion of control of composite systems by its subsystems.

The relationships between a composite system and its subsystems are seemingly simple. Once we consider the control phenomena between subsystems and component systems, a fundamental relationship of duality become clear. On one hand, the composite system trivially includes subsystems. On the other hand, the subsystem determines the full systems. This dynamic duality of acting/acted is well-modeled from the viewpoint of category theory. This provides a framework to treat the categories of morphisms of any system as entities on equal footing: small or large, subsystem or supersystem.

The relationship between systems is modeled by functors between categories of mobility. As an important example, consider the subsystem _{0}, especially_{0}. Then, we can define the notion of control as follows.

Let

One remark on the notion of a composite system is that it includes both the body of the robot and the body of the object under study.

Robots control target systems by making themselves subsystems of the composite system with the target systems. The notion of hardness/softness, which is central to soft robotics, can be defined as follows.

A robot is hard when the category of mobility of the composite system of the robot and target entity is isomorphic with the category of mobility of the robot during interaction.

We emphasize the term “isomorphic;” i.e., there is the invertible functor between them. This means a coherent one-to-one correspondence between the state and transitivity of states of the composite system and robots during interaction, as schematically shown in Figure

Mathematical understanding of the difference between hard and soft robotics. (a)

A robot is soft when the category of mobility of the composite system of the robot and target entity is categorically equivalent to the category of mobility of the robot during interaction.

We here emphasize that the difference between a hard robot and a soft robot has been successfully defined via exact mathematical notions as the difference in the sameness given by isomorphic and categorical equivalence. Consequently, the categorical equivalence can provide indeterminacy in terms of the control. In other words, as long as the categorical equivalence is satisfied, potentially multiple, in some cases huge, physical states yield the same meaning. In certain contexts, this provides much power for the finding the best or approximately the best way to control, as schematically illustrated in Figure

However, not all soft robots are effective because overly soft robots will not work for detailed control. In that sense, there is a tradeoff between finding and keeping the (approximately) best way of control. In the next subsection, we define the notion of effectively soft robots.

What kind of soft robots are effective? To reflect the tradeoff between finding and keeping, we define the notion of

A state

A category of mobility is effective if there is an arrow from any state to some critical state for the soft–hard transition.

A soft robot is effectively soft if the category of mobility contains critical states.

The notion of effectively soft robots provides a new idea for the powerfulness of soft robots.

We have focused on the universal gripper [

One answer is the vacuum machine because the bag cannot keep its best shape without its help. However, there is another factor: the size of coffee beans.

It is natural to imagine that a smaller size means that the beans have higher mobility. In other words, the category of mobility becomes rich with arrows, and the robot becomes softer. However, the critical states become scarce, so the category of mobility becomes less effective. If this reasoning is correct, there should be optimal size of coffee beans that make the universal gripper effectively soft. In the next section, we present numerical simulations performed to investigate this aspect.

In addition, the notion of categorical equivalence in the definition of a soft robot clearly conveys that fluctuations or deformations are inherently accommodated. Furthermore, in the case of the universal gripper, concerning the fact that the composite system is made by the gripper and the target, the adaptation processes will differ depending on the shape of the target to be handled. In the meantime, we will show that the number of states and transitions is indeed huge if we consider the smaller physical scale involved.

To demonstrate the autonomous adaptation of soft robots manifested by the category of mobility as well as quantitatively present the prediction and indication by the theory of effectively soft robot, we present a model system and analysis of a universal gripper [

Before proceeding, we add a few remarks regarding the use of soft gripper. As introduced above, we consider that soft gripper is one of the most visible examples in which the concept of category of mobility is directly applied. We also emphasize that many environmental conditions besides soft gripper and target objects are

The object to be grasped is depicted by a one-dimensional surface profile, as schematically shown in the lower side in Figure

Model of the autonomous adaptation with soft robotics. (a) Schematic diagram of a soft-material-based hand

Note that each particle in the gripper is not controlled individually; the particles can freely move but are subjected to such a constraint that the total volume, or total number of particles in the glove, is constant.

To highlight such a mechanism, we present the following hierarchical model. Suppose that the surface profiles of the object (i.e., TARGET) and gripper (i.e., GRIPPER) are given by

The relative difference

The flow of particles in the glove of the gripper may occur in a region where the difference between the target and gripper is more evident. To quantify such a property, we introduce the following scale- and position-dependent fitting measure:^{N}, namely, the number of pixels is given by 2^{N}. For simplicity, we can assume that^{S}, there are^{S} areas in total at the corresponding scale^{(0)}^{(0)}(2^{N}) are the height information at the finest scale (

The movement of particles between adjacent areas is autonomously induced at locations where the scale- and position-dependent metric

Calculate

Find the scale and position that maximize

Decrease the height of the corresponding area of GRIPPER by a unit if the sign of the content of equation (^{S}, are flowing out to the neighboring areas. Similarly, increase the height of the corresponding area of the GRIPPER by a unit if the sign of the content of equation (

Because of the flow of particles getting out of or getting into neighboring areas in Step ^{S-1}) go to the left side, and the other half (2^{S-1}) go to the right side. There are 2^{S} locations for the particles to be settled in the left or right areas. We randomly choose 2^{S-1} positions within such areas of the left- and right-hand-side neighbors, respectively, and reconfigure the height accordingly.

Repeat Steps

When the area that maximizes

Figure ^{10} (= 1024).

Demonstration of the scale-dependent adaptation with soft robotics. (a,b) Assumed profiles of the target object. Both (a) and (b) have the same surface roughness (_{min}), the dynamics exhibit different characteristics. (e) Estimated number of states and transitions between states at different scales of the model. (f, g) Comparison of the shape change of the soft robot during the adaptation to different target objects [(a) and (b)]. The evolution of

The degree of adaptation of GRIPPER to TARGET is evaluated by_{a}, is used as a measure to quantify the surface roughness in the literature [_{a} with respect to the profiles shown in Figures

Figure ^{S}) were given by 4, 8, 16, 32, 64, and 128. We repeated the evolution of GRIPPER 1000 times, beginning with the same initial condition (flat surface), and examined the average values in the following analysis.

As shown by the red curve in Figure _{a} decreased as time elapsed. We configured _{a} when _{a} increased with the minimum scale, which means that the fit between GRIPPER and TARGET was not perfect. Note that increasing the minimum scale obtained certain_{a} faster than at larger scales. For example, until the time cycle of 375,_{a} decreased most rapidly with _{a} decreased fastest with _{a} decreases in the same manner until around cycle 200, regardless of the given

These results are accounted for by the mathematical framework presented by the category of mobility: the richness of the category of mobility with regard to the minimum physical scale of the model (_{a}; however, this can mean that the robot is too soft, either in that the possible friction between the robot and target is weak or that it takes more time to reach a steady state (constant_{a} value). Thus, the concept of an effectively soft robot is demonstrated.

Indeed, the number of states and transitions between states in the numerical model increases exponentially as the physical scale of interests becomes smaller, as shown by the red circles and the blue squares in Figure ^{S} blocks over a horizontally expanding area of^{S}. A transition between states is defined such that a block, among^{S} blocks, moves toward either the right- or left-hand side. As shown in Figure ^{75} and 10^{77}, respectively; whereas at a coarser scale with^{3} and 10^{4}, respectively. With these calculations, it is clear that there is an overabundance of states and transitions, which was argued through natural transformation and the definition of “effectively soft robot” in the previous section. By the way, Mathematica (Version 11.3) was used in deriving the number of states and transitions.

Furthermore, the dynamics of the particle flow behaves differently depending on the shape of the object. Figure _{a} regarding TARGET A and B, which are marked by red and blue curves, respectively._{a} decreased faster with TARGET A than B. Although the initial value of_{a} was the same for A and B, they had different spatial frequencies, as shown in equation (^{5} (= 32). Specifically, here, the number of transferred particles at cycle

When the cycle was around 200, the number of moving particles decreased significantly. This corresponds to the situation where the adaptation between the target and gripper progressed only at the finer scales. This is physically natural because TARGET A had a simple periodic structure; hence, the selected physical scale was basically monotonically decreasing. In contrast, TARGET B exhibited quite different sequences. Because TARGET B had two spatial frequencies, the physical scale that maximized

Before finishing the discussions, we add a few remarks. First, while the present study specifically deals with universal gripper as a typical example, the categorical approach in the present paper has general versatile applicability to various sorts of soft robots in such forms as soft-legged robot, octopus’ arm, among others. We emphasize that our approach is surely applicable to other various platforms on the basis of the following reason.

The point is that the effective softness of a robot can be defined whenever we can define the categories of mobility of the composite systems of the robot and the targets. The composite systems in the case of universal gripper are the composite systems of the gripper and the targets to be grasped. The robot is called effectively soft if its control on any target of certain kind is effectively soft. In the case of the walking soft-legged robot, for example, we can define the effective softness by considering the composite systems of the robot and the system of certain kind on which the robot will walk.

Conversely, let us simply consider a continuously deformable soft material, such as a soft ball, under gravity. Here the notion of effective softness cannot be well defined as itself because the equivalence, namely the class of target systems, is not given. Once the class of target systems is specified, in other words, the

Finally, we discuss the contribution of the present study and future works. One may ask the following question: Is this study a design theory or an architecture for soft robots? We here state that it is the former. We consider that there are huge descriptive benefits in the design of soft robots where the essence of category theory, especially natural transformation, is greatly utilized. Meanwhile, the benefits in synthesis or optimization in the design process of soft robots are unfortunately

From these considerations, an important future study is construction of theories that provide visible and engineering insights and give fundamental limits for soft robotics from category theory. Indeed, Naruse et al. proposed “short-exact-sequence-based time” based on homological algebra and triangulated category to account for the maximum operating speed in a photon-based solution searching system [

We have proposed a mathematical foundation for soft robotics based on category theory. The category of mobility with the notions of functors and natural transformation provides a rigorous formulation for soft robots and their interactions with the target object or environment. The difference with hard robots and the effectiveness of a soft robot were mathematically described: the former and the latter are based on isomorphism and categorical equivalence, respectively. Natural transformation provides rich transitions between states providing deformable, autonomous, adaptive behavior of soft robots, whereas a one-to-one correspondence between the state and transitivity of states restricts the rigid movements of hard robots. In the meantime, overabundant transitivity could lead to unstable behavior; for example, a surface profile that is too smooth may not realize successful grasping of target objects. We have introduced the notion of effectively soft robot in order to realize intended functionality with the notion of critical state. As an application of the theory, a model system and analysis have been presented to examine the adaptation behavior observed in universal grippers. The scale dependency of the elemental particles contained in the gripper was observed to agree with the theoretical prediction; indeed, including an overly small scale yields slower adaptation. It should be noted that the scale-dependent adaptation measure has been inspired by the notion of effectively soft robot in the theory. The number of states and transitions therein has been numerically estimated, where smaller scale yield huge numbers. The autonomous adaptation behavior has been demonstrated where different state transitions were clearly observed depending on the profiles of the object where even oscillatory behavior has been observed.

The power of category theory in its descriptive aspect is clear. The benefits in synthesis and optimization have been partially demonstrated in the present study through the modeling and analysis of the universal gripper. As discussed at the end of Section

The data used to support the findings of this study are available from the corresponding author upon request.

The authors declare that there are no conflicts of interest regarding the publication of this paper.

We acknowledge K. Nakajima for variable discussions regarding theories and implementations of soft robots. This work was supported in part by the Core-to-Core Program A. Advanced Research Networks, Grants-in-Aid for Scientific Research (A) (JP17H01277) from Japan Society for the Promotion of Science, and CREST program (JPMJCR17N2) from Japan Science and Technology Agency.