Fan Xu, and Hesheng Wang,

Abstract—Robotics has aroused huge attention since the 1950s.Irrespective of the uniqueness that industrial applications exhibit,conventional rigid robots have displayed noticeable limitations,particularly in safe cooperation as well as with environmental adaption. Accordingly, scientists have shifted their focus on soft robotics to apply this type of robots more effectively in unstructured environments. For decades, they have been committed to exploring sub-fields of soft robotics (e.g., cuttingedge techniques in design and fabrication, accurate modeling, as well as advanced control algorithms). Although scientists have made many different efforts, they share the common goal of enhancing applicability. The presented paper aims to brief the progress of soft robotic research for readers interested in this field, and clarify how an appropriate control algorithm can be produced for soft robots with specific morphologies. This paper,instead of enumerating existing modeling or control methods of a certain soft robot prototype, interprets for the relationship between morphology and morphology-dependent motion strategy,attempts to delve into the common issues in a particular class of soft robots, and elucidates a generic solution to enhance their performance.

I. INTRODUCTION

THE term of “soft robotics” is primarily defined as an autonomous system with rigid mechanism and flexible joints that exhibits passive compliance, or a system made from low modulus materials, characterized by inherent compliance from elastic/hyper-elastic materials. In this paper, the latter definition is highlighted to present the progress in this area,which covers the advanced modeling and control methods that have been practically applied in non-industrial environments.Soft robotics is considered a novel branch of robotics, which is an intersection of multiple disciplines, e.g., biology,chemistry, electromechanics, and electrochemistry, etc. It has generated rising interest for its potential to induce seamless interaction between robots, human beings, as well as the environment. It has also been known for its ability to eliminate the unintentional collision and impact with the coexistence between robots and humans. Conversely,conventional rigid robots have displayed significant limitations in addressing this issue [1], [2]. Several comprehensive review papers have been published [3];moreover, several narrative review papers have been made,primarily reviewing fabrication [4] and actuation methods(e.g., fluid-driven actuation [5], [6], shape memory alloy(SMA) [7], dielectric elastomer-based actuation (DEA) [8],[9], granular jamming actuation [10], actuation for small-scale robots [11]), control algorithms [12], and applications (e.g.,biomedical use [13]–[15], biomimicry [16], [17]).

In contrast with the mentioned review papers, this paper aims to develop a framework to determine how to mathematically model a type of soft robot prototypes;subsequently, we discuss appropriate control algorithms based on the specific morphology and characteristic of this congener to enable the soft robot to perform in target control tasks. In this regard, we delve into the underlying principle of the motion of most existing soft robots that follow different mechanisms to clarify a strategy to unify modeling, sensing,and control methods into a certain type of soft robot for readers interested in relevant fields. To this end, the paper enumerates typical morphologies and configurations of existing soft robots, and then presents unified technologies of modeling, sensing and control algorithms with typical cases.Hopefully, this paper can outline feasible methods according to different morphologies and tackle common issues facing the respective type of soft robot.

It is known that soft robots made of compliant materials outclass traditional rigid robots in cooperation and coexistence with humans, undertaking tasks without being given absolute environment information, locomotion in rough terrains, etc.They are also known to be advantageous for their capacity in tasks that rigid robots cannot deal with [18]. For the mentioned properties, soft robotics is considered a solution that boosts the evolution of robots in unstructured environments [2]. In the meantime, however, challenges facing the actual implementation of soft robots are generated,which can be summarized into four aspects in the relevant scientific literature [3]:

1) Fabrication of its unique mechanism;

2) Accurate mathematical model of the system;

3) Intrinsic sensing device that can be utilized for the deformable mechanism;

4) Real-time control algorithm to ensure vigorous performance.

Over decades researchers have committed to the soft robotics related branches (e.g., advanced material investigation, smart actuator design, compliant sensing device development, modeling, and control [19]–[22]) to terminally enhance the performance of soft robot system.

In the following, several common types of soft robots will be exemplified, and corresponding modeling and control strategies that are feasible for such specific types are elaborated to provide a new researcher with an intuitive framework. Biology (e.g., human beings) is commonly considered to be highly adaptable to uncertain environments for their high-level sensing systems and sensorimotor control.Researchers have fixed their concentration on biology to find something that can be implemented into soft robots. To this end, by reviewing numerous studies, it is found that existing soft robots usually engage in similar morphology and motion modes with living biology. Prevalent morphologies usually adopted in soft robot systems will be enumerated. Three types of soft robots are reviewed, which include the soft continuum robot (Section II), soft gripper (Section III), and soft mobile robot (Section IV). Each section exemplifies the current progress on modeling and control methods and ends with a short discussion. Section V summarizes this paper and provides a conclusion of the existing approaches, technical limitations and potential study trends in the domain of soft robots. The outline is demonstrated in Fig.1.

Fig. 1. Outline of the structure.

II. SOFT CONTINUUM ROBOT

A soft continuum robot can be considered as the intersection of a soft robots and continuum robots. Continuum robotics has become a flourishing research area with various work on its design, modeling, control, and applications. Partial continuum robots are made of rigid materials, distinguished by a higher young’s modulus as well as higher inherent stiffness [3]; yet soft continuum robots are more compliant and characterized by continuous deformability of their backbone, thereby exhibiting infinite degrees of freedom. Compared to rigid continuum robots which are usually actuated by embedded tendons and can achieve bending and twisting deformation(the latter depends on the accommodations of tendon routines), soft-bodied robots demonstrate more flexibility in deformation; besides bending and twisting, they are sometimes also capable of stretching and contraction when adopting pneumatic actuator. Due to better deformability, soft continuum robots exhibit better shape adaptability but are not satisfying in load capacity or stiffness. A comparison between these two types of robots is given in Table I.

A. Morphology

Invertebrates, exemplified by the octopus, are characterized by impressive dexterity and vigorous grasping behavior, even though they do not have a spine. Researchers have been interested in exploring its structure, neural system and the underlying principles of sensorimotor control to reproduce their powerful behaviors on the robot platform. Other than investigating whole invertebrates, some researchers have concentrated on investigating invertebrates’ appendages.Octopus tentacles, demonstrating large-ratio extension and contraction ability, can squeeze themselves into a highly constrained environment, reach and fetch the object rapidly during predation. Soft continuum robots that are based on the morphology of octopus tentacles, trunk and other analogous morphologies of the living organism, are one of the primary types of soft robots and have been continuously growing over the past decades. We will exemplify studies of soft continuum robots, except for those that are rigid ones, in this section.

Biologists have reported on the special structure of invertebrates that enables their powerful sensorimotor abilities, which is the mechanism composed of muscular hydrostats. Such structure shows astonishingly accurate controllability of each piece of muscular tissue to support diverse motions (e.g., elongation for reaching, contraction for squeezing into a small room, as well as omnidirectional bending for dexterous behavior). The studies on the characteristics of such unique tissue that enable distinct performance have long been conducted [23]–[26]. Though the theoretical model to describe the motion of octopus tentacles is developed, such motion is yet unlikely to be thoroughly reproduced in a robot system. Difficulties may originate from several limitations of techniques in its development include the design of controller to mimic the neural system, feasible sensing devices to provide considerable environmental information, etc. In brief, there remains a gap between the theoretical analysis of biomechanics and how it is adopted in the robot systems to reproduce the identical advanced and intelligent sensorimotor control. The breakthroughs of soft robotic techniques require multi-discipline contributions.

Numerous research has presented soft robot prototypes that exhibit analogous morphologies to mechanisms of living biology. In many publications, different mechanism designs to mimic the shape and function of the octopus tentacle[26]–[31] and trunk [32]–[39] have been found. Figs. 2 and 3 demonstrate several soft continuum robot prototypes. The proposed robot prototypes permit a motion pattern that can perform bending [28], [40]–[47], and sometimes elongating and contracting motions [26], [29], [32]–[39], [48]–[51]. In[46], a soft manipulator outfitted with a pneumatic actuation method is fabricated and employed in a grasp-and-place taskby adopting open-loop control strategies. A similar mechanical structure of this manipulator has also been employed in several publications [40], [41], [48]. A redundant octopus-like soft robot manipulator is presented in [29], [31],[52], [53], consisting of three pneumatically actuated sections accommodated in cascade. Each of the sections is composed of three parallel-accommodated chambers with periphery fiber to constrain the expansion radially and enable unidirectional extension axially. Accordingly, the bending and elongation performance of the manipulator can be obtained by appropriately regulating the synergy motion of these bellowlike chambers. In [30], EGaIn-based soft sensors are integrated with the pneumatically-actuated soft robotic manipulator to acquire the motion state and enable vigorous grasping.

TABLE I COMPARISON IN RIGID CONTINUUM ROBOT AND SOFT CONTINUUM ROBOT

Fig. 2. Soft continuum robot prototypes inspired from octopus tentacle. (a)12-cable-driven soft robot arm [27] (Copyright ? 2014, IEEE); (b) Bioinspired manipulator with hybrid tendon and pressure actuation [28](Copyright ? 2015, IEEE); (c) OctArm V [29] (Copyright ? 2008, IEEE);(d) Pneumatically actuated soft robotic tentacle with integrated strain sensors[30] (Copyright ? 2020, IEEE).

It is noteworthy that the time delay and hysteresis may be caused during pressurization and vacuum process, and the control performance will be consequently degraded. Such effect can be reduced by the presented parallel-accommodated independent actuator [32], [33]. A similar actuation method has been extensively adopted in existing studies [34]–[39].Other fluid-driven soft continuum robots, as presented in several studies [48], [54], have been deeply investigated in terms of their modeling and control strategy. Similar morphology of an octopus tentacle or trunk can also be found in [28], [50], [55]–[58]. A silicone-made cable-driven soft robot manipulator is cast into a similar shape with an octopus tentacle. Omnidirectional bending performance can be achieved by pulling the embedded cables to regulate the orientation of bending as well as the bending angle. Maghooaet al. [28] presented an octopus-inspired soft manipulator using a hybrid actuation method. The proposed prototype is characterized by multi-bending performance, as enabled by regulating tendon displacement, and variant stiffness under pressurization and depressurization of the pneumatic actuator.A similar actuation method was also adopted in [59] to obtain double functions of both accurate positioning and adjustable stiffness. Such hybrid actuation methods have been proved to be feasible to actively control the stiffness to achieve a wider range of load capacity [60].

B. Modeling for Soft Continuum Robot

In this section, we review commonly adopted kinematic and dynamic modeling methods and summarize their characteristics in Table II.

Fig. 3. Soft continuum robot prototypes inspired from trunk. (a) Festo’s Bionic Handling Assistant with pneumatically actuated bellows [34](Copyright ? 2017, IEEE); (b) Pneumatically actuated soft manipulator with reinforcement learning closed-loop control [51] (Copyright ? 2019, IEEE);(c) Pneumatic muscle based continuum robot with embedded tendons [32](Copyright ? 2018, IEEE); (d) Trunk-like arm with 12 fluid actuators and 12 proprioceptive sensors [58] (Copyright ? 2020, IEEE).

1) Kinematics

Soft continuum robot manipulators display similar mechanical properties to rigid continuum robots that consist of cascaded rigid joints, which have aroused increasing attention over decades [61]. Researchers have delved into their mechanism and motion pattern and have also committed to accurately modeling the system and designing closed-loop controllers to enable the robot a better applicability. Despite the different characteristics between rigid and soft continuum robots, the analogous principle of motion of both is assumed(as listed in Table I). In this view, the modeling and controlling methods can be considered versatile in these two domains. Thus, the modeling methods studied in domains of both rigid and soft continuum robots have been reviewed. As a result, benefiting from similar kinematics, modeling work is tractable by referring to the modeling method of a rigid continuum robot. The universal modeling and control method in a continuum robot will be reviewed here whether it is made from a soft or rigid material. Unlike conventional rigid robots with articulated prismatic and revolute joints, those soft robots are characterized by continuous deformability along the backbone at an arbitrary position. Hence, they are considered as systems with infinite degrees of freedom (DOFs) in the configuration space. In most cases, it is assumed that soft robots exhibit a hyper redundant mechanism that permits high but finite DOFs when being kinematically modeled.

There is a method adopted extensively to model this type of robot manipulator, which is the so-called constant-curvature based kinematic modeling method [62]. Such a method differentiates that of conventional rigid robots with prismatic and revolute joints in their depiction of configuration space.On the whole, this method is built on the hypothesis that the shape of the backbone after its deformation can be considered to consist of sequential arcs, each of which exhibits invariant curvature. Subsequently, the spatial configuration parameters are then defined to depict the curve of the backbone in the form of the shape consisting of cascaded arcs. When a continuum backbone is thought to meet the constant curvature assumption, the modified Denavit-Hartenberg (DH)convention can be computed based on arc geometry. Frenet frame is introduced to describe such a quasi-circular shape in the robot configuration space based on artificially defined generalized coordinates. Thus, one can compute local transformation between two adjacent robot sections and then express the forward kinematics abiding by the chain rule. This method is well accepted in modeling congeneric soft continuum robots with planar or spatial motion [26], [28],[36], [40], [42]–[49]. Recently, it has been verified that this method can also be used to model the self-growing soft robot[63]. The modeling method based on constant curvature assumption is known as an analytical solution since it leads to an explicit expression of the system kinematics, thereby contributing to model-based control. Despite the demonstration of potential and feasibility to the employment in closed-loop controller design, it is argued to be less accurate especially when external loads are applied on the soft robot. In such cases, the section will not remain the same curvature. This can be addressed by adopting variable curvature assumption [33], [34], [36], [37], [56]. By discretizing the continuum backbone into multiple virtual sections, each virtual section can then be considered to permit a constant curvature while a variable curvature is compared with that of its adjoining virtual section. By enhancing the discretization of the soft continuum robot, the constantcurvature constraints can be maintained in each virtual subsection and the model accuracy will be optimized in comparison with the physical model. This modeling method can generally depict the circular-like configuration of a soft continuum robot and construct the mapping,findependent, of the configuration space and Cartesian space, while specific mapping,fspecific, of the actuation space and configuration space should be solved. For instance, when the tendon-driving actuation method is adopted, the configuration parameters,which are arc length, orientation angle and curvature radius,can be functioned with respect to tendon variables [56], and the forward kinematic mappingfspecificcan be consequently constructed. The kinematics from the actuation space to the Cartesian space can be obtained by combiningfspecificwithfindependent, as seen in Table II.

Besides analytical solutions to kinematics by mathematically modeling the system, numerous studies also focused on numerical modeling methods and model-learning techniques to solve system forward and inverse kinematics[64]. [65] presented a framework of kinematic modeling method for continuum robots using the modal approach,where the spatial curve shape of the continuum robot was depicted in terms of the polynomial function. Digumartiet al.[66] have also employed shape function to model the shape ofthe continuum robot and express the configuration in the form of Fourier’s series. By using such a method to model a continuum robot, numerical optimization methods are usually adopted to solve the curve-fitting problem and determine the specific shape function best approximating the spatial curve of a continuum robot. Modeling learning methods have been studied, contributing to learning-based open-loop control [51],[64], [67]–[73]. To learn an accurate system model,considerable motion data of system input and corresponding output have been collected to span the training set. Obviously,the accuracy is highly dependent of the feasibility of the training set. Goal babbling method is proved to be viable in generating sufficient training data and addressing the multiple-solution problem in inverse kinematics of the hyper redundant manipulator [69]. The obtained inverse kinematics of the presented soft robot then illustrates its accuracy in point-to-point open-loop control tasks.

TABLE II MODELING STRATEGIES FOR SOFT ROBOTS AND THEIR PROPERTIES

2) Statics and Dynamics

Kinematics is subject to increasing errors in the situation with unneglectable external loads and environmental contacts.Unlike rigid robots, soft robots demonstrate strong coupling between kinematics and statics/dynamics owing to the deformability of their compliant mechanism, making it less accurate to express the motion with respect to given position signals as actuation inputs. In such scenarios, the configuration is determined by the input position signal as well as the external force/torques. To be more specific, the variable geometric parameters (e.g., length of the link) due to applied forces/torques will be obtained, and then, kinematic parameters should be updated based on the strain analysis,with given force conditions. Hence, to accurately model the system in the situation with existing external loads or interactive effects, the variance of kinematic parameters caused by imposed forces and torques should be highlighted,and a feasible way should be developed to eliminate their effects. To this end, the coupling mechanical properties should be studied to achieve more sophisticated and accurate system model. Researchers have been committed to the study of statics [52], [55], [74]–[76] and dynamics [27], [50], [54],[77]–[82] of the continuum soft robot. It is known as a prerequisite to enable a force control strategy or accelerationbased control strategy. By discretizing the continuum robot into multiple virtual sections, the motion of each of them can be considered to integrate several prismatic and revolute rigid robot links accommodated sequentially. Accordingly, subject to constant curvature or variable curvature assumption,dynamic modeling methods for conventional rigid robots with articulated joints can also be verified in continuum robots. The Lagrangian formulation and Euler-Newton equation, which are among the most frequently adopted methods in modeling rigid robotics, are also feasible in this case. It is noted that kinematics is a prerequisite under which the position and velocity of the center of mass can be computed. Afterwards,the mechanics or statics of each virtual section can be formulated based on force, momentum or energy equilibrium.By combining all equilibriums for each discrete virtual section, a normalized form of system mechanics or statics is obtained. The internal energy change due to the deformation of the mechanism should be considered. It can be modeled based on the constitutive law of materials characterized by hyperelasticity or viscoelasticity [83]. Marcheseet al. [40]presented a discrete modeling method to express the dynamics of the pneumatic soft manipulator in a similar shape to trunk.This method is also known as an analytical method to model the system since the closed-loop form of motion equations can be obtained this way. Next, the solved model proved its accuracy in open-loop point-to-point control tasks. A similar analytical method has been extensively adopted in soft continuum robots. Reference [54] presented a discrete method to model the system leveraging the Lagrange formulation. The presented soft robot arm was in a similar shape to that in [40]consisting of three cascade sections, each of which has three independent bellow-like chambers. The motion of this manipulator is obtained by the generated longitude force from pressurization of each bellow. The specific bending and elongation performance can be achieved by appropriately adjusting the length of each chamber. In their studies, the actuation space inputs, which are pressures generated by the air pump, can be accurately mapped to the robot motion of bending and elongation in work space. Thus, the performance of positioning and tracking can be achieved by regulating the input pressure to obtained reference force driving actuators to deform to the desired pose. Nevertheless, it is argued that such a method shows constraints in situations where external forces, e.g., gravity, contacts, etc., exist and enhance the configuration coupling effect. In the mentioned scenarios, the perfect circular shape hardly permits pure bending motion hardly permits. Thus, augmented generalized coordinates are defined to better capture the spatial shape of a continuum robot with torsion and contraction along its backbone [50],[84]. This modeling method has also proved its feasibility in aquatic environments where fluid interactions are considered[80] to extend its application into underwater environments[85].

Apart from the aforementioned analytical solutions based on the discretization assumption, numerical solutions based on continuum methods, are also comprehensively studied.Continuum methods originate from beam theory, which is considered a branch of solid mechanics and exemplified by Cosserat rod theory for the spatial-motion case and Euler-Bernoulli beam theory planar-motion case. Highly coupled partial differential equations (PDEs) construct the distributedparameter model and depict the motion of a continuous curve in the Frenet-Serret frame. The statics can be expressed as a set of ordinary differential equations where the variables are considered time-invariant, while dynamic motion equations comply with the rule represented by PDEs where the variables are both space- and time-variant. To better understand the underlying principle of this method, interested readers can refer to [52], [53], [86], [87]. The continuum method is characterized by better handling of the coupling between kinematic parameters and applied loads. The stress analysis is formulated on the infinitesimal segment, and is assumed to be rigid. Given the constitutive law of elastic or viscoelastic material, the linear strain and angular strain can be modeled into the expression consisting of terms of the linear/angular velocity, acceleration, and external wrench. Note that the linear and angular strain can be formulated as a partial differential of the position vector and Euler angle with respect to arc length, while the velocity and acceleration exhibit a differential and doubly differential form with respect to time.Accordingly, by solving the set of PDEs numerically, robot configuration at arbitrary position can be obtained along the continuum robot backbone. Trivediet al. [29] have developed a geometrically exact model based on the Cosserat rod theory for the soft robot manipulator, OctArm V. The model exhibits high accuracy even if a large deformation of the robot is observed, given consideration to nonlinear strain model.Though they present the dynamic equations in their study, the validation concentrates on the static case. Rendaet al. [77]also leveraged Cosserat rod theory to model system dynamics of an octopus-tentacle-like soft manipulator. In their paper, a general method was proposed to model the system dynamics of the congener of cable/tendon-driven soft continuum robots.Unlike that of fluid-driven soft robots aforementioned [29],they additionally considered the contact along the cable/tendon path as evenly distributed force and modeled in line with centripetal force [73]. Subsequently, they extended the system dynamics into an underwater environment by further considering hydrodynamics [27]. Dynamic modeling work on congeneric cable/tendon-driven continuum robots can also be found in [78], where external loads were considered.While the continuum methods are reported to be more accurate in modeling the system motion equation under a large deformation or deflection, it is subject to the high computation load hindering the implementation in real-time control. This is because there is no explicit expression of the Jacobian matrix, and its numerical solution requires tedious iterative computation and extra work to solve the boundary and initial conditions. To this end, Ruckeret al. [88] proposed a feasible way to calculate the Jacobian when modeling the system with a numerical method, in the scenario with external loads. To enhance the ability of real-time computing,researchers have contributed to improving computation efficiency. Tillet al. [89] first presented the system dynamics of the continuum robot based on Cosserat rod theory and then proposed a real-time computing method to the numerical solution to the model based on time discretization.Accordingly, the problem was simplified to find the solution of ordinary differential equations (ODEs) every time step.Giorelliet al. [72] further investigated the solution to inverse statics based on the iteratively solved Jacobian for a planarmotion soft robot. Subsequently, they verified its accuracy by implementing it into a feedforward force control. Finite element model (FEM)-based methods also require real-time computing for solving the updated node positions in every time step. In primary cases, system statics are modeled based on the force equilibrium. Solving the compliance matrix is considered the critical point to this modeling method [90],[91], based on which the displacement of actuator and the system states can be computed by given external forces.Considering the coupling effect, FEM-based method can enable a position control performance [92]. Recently,Graziosoet al. [93] demonstrated a unified method to solve system dynamics, inverse kinematics and differential kinematics by combining the Cosserat rod theory with the helical shape function based FEM.

A learning-based method is reported as a feasible method to robotic modeling and control without any pre-knowledge of a robot system. Researchers have studied the implementation of neural networks in soft robot modeling work [71], [72]. In the study, the inverse statics learning method was developed based on a feed-forward neural network. Subsequently, openloop control was adopted to validate the model performance in the two-dimensional positioning task. Despite the simplification in mathematical modeling work, modellearning methods cannot permit feedback control due to paucity of an analytical solution to the Jacobian that can accurately build the map between the robot actuation space and workspace. It is known that with the existence of a Jacobian matrix, the reference inputs can be computed as per the measured states and output feedback respective sampling time. For the system adopting model learning method, little knowledge on how kinematic and dynamic parameters affect the system performance is available, making it difficult to iteratively modify the inputs according to the error feedback.Therefore, model learning methods have usually been employed in open-loop control tasks, the accuracy of which is highly dependent on the training process. These methods result in no solution to address possible modeling errors that can be done in the case of closed-loop control with knowledge of the system. Consequently, in such system, the divergence of output error is hard to theoretically prove to warrant system stability. Besides the case of closed-loop control-mentioned general modeling methods, specific characteristics (resulting from different actuators, materials, and configurations) should also be considered. An accurate model to depict deformation is required. There are numerous mathematical methods that contribute to modeling nonlinear materials with the characteristics of elasticity, hyperelasticity or viscoelasticity.To be specific, the Ogden model, the Yeoh model, and the Polynomial model are frequently employed in modeling the stress and strain [83], [94], [95]. For instance, the Neo-Hookean model was adopted in [29] to formulate the constitutive relation of the material and model the nonlinear axial strain energy. The cable/tendon-driven actuation method exhibits compactness, rapid and stable force-transmission,which has been well studied and extensively employed in medical and surgery use [96]. Despite the fact that this method has its benefits, it suffers from unneglectable friction along the tendon path, degrading accuracy in the control task if no solution is created to address these issues. Though such effect can, to some extent, be mitigated by properly designing sleeves, which can enclose the embedded cable/tendon and avoid direct contact between the cable/tendon and the soft robot, the friction effect remains and causes a loss in transmission force along the cable/tendon path [97].Subramaniet al. [98] proposed a mathematical model to express the friction effect in a congeneric cable/tendon-driven continuum robot. It is also reported that such an effect could be compensated by model-based parameter identification. Royet al. [99] assumed an exactly known model of statics with unknown parameters of friction, and then optimized the parameters by minimizing the mismatch in measured and computed tendon variables, as well as input forces.Furthermore, the extension of the cable/tendon due to elasticity also results in inaccuracy. Camarilloet al. [100]presented unified mechanics that consider such effects.Subsequently, the model was implemented in the tracking task in robot configuration space [101]. Therein, reference tendon inputs were computable if given the desired configuration with a solved system model. For soft continuum robots with advanced materials as their actuators (e.g., SMA [7], DEA [8],[9], etc.), unneglectable hysteresis phenomena shall generate interest. Researchers tend to adopt mathematical models to compensate for nonlinear hysteresis effects (e.g., Preisach model [102], Prandtl–Ishlinskii model [103], etc.) to further reduce modeling error and to enhance performance in control[104]. Here we only present clues for different mathematical modeling methods to solve hysteresis rather than provided detailed explanations of them. For more information,interested readers can refer to the reviewed paper preceding.

C. Control Strategies for Soft Continuum Robot

With the solved system model, a specifically designed controller can enable desired performance of a soft robot in certain tasks. Usually, closed-loop control is preferable to an open-loop strategy in positioning and tracking tasks requiring high accuracy. In this section, we review existing control strategies of soft robots and summarize their characteristics,strengths/weakness in Table III, followed by sketches of control architectures (including model-based kinematic/static control (Fig. 4), dynamic control (Fig. 5), and model-less control (Fig. 6)) and open-loop control architecture (Fig. 7).

1) Kinematic/static Control

It is known that the analytical solution of a system model can contribute to a closed-loop control algorithm with appropriate selection of different intrinsic or exterior sensors that can provide shape [105]–[110], position [48], [111],[112], tactile and other feedback information [113], [114].Palmeret al. [115] demonstrated the model-based configuration trajectory tracking performance of a continuum robot with a similar shape to a snake. In their study, shape tracking control and end effector positioning were simultaneously permitted. Gilbertet al. [116] investigated a similar problem in the case where pre-curvatures exist and the realized leader-following motion synchronizes with insertion into constrained terrain. Their studies further enhanced the potentials of a continuum robot in the application of in vivo navigation in minimally invasive surgery [117], [118]. A graphical measurement system is known for its high economy and its ability to provide considerable environmental information. It has been broadly employed for shape sensing[105] and visual servoing control [85], [112], [119].Researchers have studied vision-based control and have been committed to specific problems. To solve the unknown camera mapping between the image space and robot inputs with an uncalibrated camera, Fanget al. [119] proposed an online method to solve the camera model based on Gaussian process regression, and enabled the soft robot to execute positioning and tracking tasks. Uncalibrated problem can alsobe addressed using adaptive method [85], [120]. In these studies, online estimation laws were designed to update unknown camera parameters. Appropriate control algorithms were also proposed to enable kinematic visual servoing control [120], [121] as well as dynamic visual servoing tracking control [85]. It is known that kinematic controllers are highly restricted to control tasks with environmental interactions. The mapping between a configuration space and actuation space will be less accurate due to uncertain interactive effects, including the contact with the external environment, unmeasurable forces exerted on the system, and other disturbances. To this end, interaction controllability is also analyzed in continuum robots [39], [120], [122]–[129].An adaptive visual servoing controller was proposed in [120].Therein, the proposed adaptive law can estimate the location of contact that causes part of the cable-driven robot to be locked, after which the kinematic model can be updated. Yipet al. [129] presented an optimization-based method to iteratively solve the Jacobian matrix online by minimizing the task space error and enabled two-dimensional positioning control in the environment with obstacles. Motion control with contact was also studied in [130]–[132], where the contact model was introduced to map the external wrench in task space to the reference displacement based on environment stiffness. Liet al. [32] presented a model-less method to cope with the modeling inaccuracy. In their study,they treated entries in the Jacobian matrix as system states and estimated them using the adaptive Kalman filter. Thus, the Jacobian matrix could be solved without dependency on preknowledge of the system model but was subject to the

constraints at the small perturbation assumptions during each sampling time. The extended Kalman filter is also reported to be feasible in shape estimation [133]. Atakaet al. [134]proposed an obstacle-avoiding method. The repulsive force was modeled inspired from electromagnetic effect and was used to guide the continuum robot away from obstacles.Meanwhile, point-to-point control of its end effector could be realized. This method differentiates from other obstacleavoiding methods in that it requires no pre-knowledge of the environment [135].

TABLE III CONTROL METHODS FOR SOFT ROBOT

Fig. 4. Block diagram of closed-loop kinematic/static control.

Fig. 5. Block diagram of closed-loop dynamic control.

Fig. 6. Block diagram of (model learning-based (top) and Jacobian estimation-based (bottom)) model-less control.

Fig. 7. Block diagram of open-loop control.

It is noteworthy that with the solved kinematics, statics or dynamics of the robot, the shape [136]–[138], external loads[139]–[146], and contact [147]–[150] are intrinsically estimable, enhancing the ability of a continuum robot to interact with surroundings by optimizing its perception ability in the absence of bulky forces or tactile sensors. Over the past several years, numerous research has been conducted concerning contact detection and localization. Bajoet al.[148] proposed a novel kinematic-based framework for collision detection by monitoring the screw motion deviation,fixed centrode deviation, as well as joint force deviation[149]. Several studies have combined shape information and the kinematic/static model to estimate external forces. Xuet al. [141] proposed a solution of model-based estimation of the external load acting on the robot end effector with measured input tension on driven tendons. In the paper, the kinematics was obtained by solving arc geometry, and the statics was formulated based on the principle of conservation of energy. With the solved Jacobian matrix and ascertained cable tensions, the unknown external load could be easily computed. Khoshnamet al. [139], [150] also proposed a shape-based force estimation method to solve the force sensing problem in applications of minimally invasive surgery and biopsy. Fiber-Bragg-gratings (FBG) sensor is one of a feasible way to provide shape information [143]. Other studies on force estimation method based on system statics are fully investigated in [144]–[146]. Apart from the aforementioned methods, Royet al. [142] further studied the dynamics of the system and derived the external force expression with the consideration of dynamic effects. Interaction controllers[122]–[124], ensuring compliance of the robot with respect to the environment, were then developed leveraging intrinsic force measuring method.

2) Dynamic Control

Model-based closed-loop control strategies can generally achieve trajectory tracking performance in the task space with analytically solved dynamics of the system [34], [47], [50],[51], [80], [151], [152]. Trajectory tracking performance was obtained based on solved system dynamics using an open-loop control strategy [153], as well as following a closed-loop image-based visual servoing control strategy [154]. Note that for the dynamic control of a fluid-driven soft robot, there are usually two-stage controllers required, namely the bottom motion controller that maps the outputs in task space to the reference inputs in robot joint space, and the upper-level one that computes the controls in actuation space based on the joint space reference, respectively. Hence, the accuracy of the proposed control algorithm is highly dependent on the accuracy of modeling of these two mappings. Researchers have suggested several methods to improve modeling accuracy. Parameter identification is presented to compute the unknown system coefficients [155]. The constants in the mechanics of the pump used to pressurize the chambers were solvable. Then, a backstepping control algorithm was employed to generate reference inputs of fluid actuators with given measurements in the robot task space. In several studies,similar methods of model-based parameter identification were adopted [42], [54], [155], [156].

D. Discussion on Soft Continuum Robot

It is worth noting that in this section we review several publications presenting continuum robots with rigid parts, the major topics of which are solutions to modeling and control problems. Due to similar morphology and similar motion principles, the methods are considered universal in these two domains. The level of study on rigid continuum robots is more mature. Thus, researchers can solve common issues by referring to existing studies on rigid continuum robots and considering typical problems raised by specific characteristics of soft robots.

Soft continuum robots are usually endowed with a simple mechanism, making their motions predictable from a mathematical aspect; thus, numerous studies concentrate on the real-time modeling and control methods. Despite diverse actuation methods adoptable, e.g., fluid-driven, tendon-driven,SMA-based actuation methods and so forth, a similar circularshape motion mode of a soft continuum robot can be achieved, making it possible to use a unified method to construct the mapping between the configuration space and the task space, and to mathematically model system motion equations. Discretization of the continuum mechanism is one of the commonly employed strategies, based on which lumped-parameter model can be established. The model then serves as a basis for model-based controller design. The kinematics and dynamics based on this idea have a closedform solution similar to that of a traditional rigid robot. As shown in Table II, the sketch is given under the classification of lumped-parameter modeling, demonstrating the unified discrete method to depict the circular shape in the configuration space. The lumped-parameter model has better real-time computing capacity and shows better controloriented ability than the distributed-parameter model, which is solved based on continuum method leveraging the elastic rod theory (as shown in Table II). Yet the former suffers from relatively high modeling errors/ uncertainties to the continuum modeling method, FEM, and model-learning method,implying that error compensation measures should be taken in the upcoming controller design to avoid degrading performance. Soft continuum robots usually enjoy regular morphology and are endowed with high-level adaptability to the surroundings. Hitherto many studies have focused on theoretical modeling and control methodology. Nevertheless,unsolved issues that hinder powerful execution in real applications still exist, demanding more intensive study. For instance, an unfathomed paradox between the level of accuracy and real-time computing ability in modeling and control remains unsettled. In addition, despite progress in both kinematic and dynamic control, limited work concentrates on interaction or force control problems. What’s more, this type of soft robot features higher degrees of freedom in its configuration space than in the actuation and task space; thus it can be regarded as a redundant system (from configuration space to task space) and underactuated system (from actuation space to configuration space). This uniqueness obviously generates challenges in control tasks. A high-performance controller should simultaneously cope with difficulties coming from both underactuated and overactuated systems, e.g., the null-space control problem in an underactuated system and redundancy allocation in an overactuated system, etc. Despite numerous control strategies that have been described, few of them pinpoint this issue.

III. SOFT GRIPPER

Soft robots are uniquely qualified for manipulation tasks involving fragile objects, with which rigid robots are not competent. There is a large role for grasping with rigid grippers in the industrial environment. However, rigid grippers display constraints in daily applications, where the requirement of safe manipulation without damaging the object hinders its implementation in subtle manipulation tasks. A considerable number of methods are proposed for rigid gripper control. Visual classification of the objects after which suitable grasping poses are determined is one of the most conventional methods that have been reported in numerous publications. It is a necessity for a rigid gripper to obtain an optimized grasping pose and act at the appropriate position on the object according to its center of mass. During its operation, a point-contact model is usually developed in the task, which should be carefully analyzed to ensure a force closure for successful grasping. To be specific, rigid grippers demonstrate superiority in industrial applications due to their high precision and load capacity produced by program-byteaching motion and high-gain control loop. Conversely, these grippers lose flexibility and compliance in unstructured environments where human-machine interaction exists. Soft graspers, in contrast to their rigid counterpart, perform vigorous grasping without any requirement for pre-knowledge of the object shape. Heuristically, face or line contact, usually generated in soft gripper operation, can achieve more robust grasping performance since such contact can easily form a force closure, which is a prerequisite for a successful grasp. A soft gripper outperforms its rigid counterpart in dexterity and safety when executing tasks in daily applications due to its shape adaptability inherent from its compliant mechanism.Most existing publications specialize in novel designs that guarantee robust grasping performance. Theoretical system models of the reported soft gripper prototypes are seldom possibly attributing to the common metrics in a task not being the accuracy but the completeness of a grasping motion.Instead, data-based and unsupervised learning methods are implemented to fulfill a specified task. The following section will review the common types of soft robot grippers.

A. Morphology

The study of soft gripper continues to be ongoing with many open solutions. It is acknowledged that in terms of an industrial robot hand, a pre-knowledge of the objects is one of the prerequisites in grasping tasks to solve the planning problem and obtain a feasible grasping pose that can result in robust grasping behavior. Soft grippers demonstrate superiority compared with rigid robot grippers, especially in that they enjoy better shape adaptability, significantly simplifying the grasping process by eliminating the detection of objects. Several typical soft gripper prototypes are shown in Fig. 8. Soft robot grippers possess diverse morphologies,operate by leveraging advanced soft actuators, and are endowed with dexterity from their bio-inspired mechanism design and sometimes intrinsic sensing ability [157]–[177]. In particular, they outclass rigid robot grippers in manipulation tasks with fragile objects [178], [179]. Interestingly, there is an overlap between soft continuum robots and soft grippers,implying that similar modeling methods used in soft continuum robots can be adopted in soft grippers. Note that when referring to the terminology of a gripper, we assume that a gripper sometimes includes the mechanical design of multiple fingers, each of which usually permits continuum properties. In this light, the difference between this section and the previous section is that in the study of grippers,emphasis will be laid on the synergy motion strategy rather than the specific motion of a single continuum soft finger.

Fig. 8. Soft gripper prototypes. (a) Soft robotic gripper with passive particle jamming [157] (Copyright ? 2017, IEEE); (b) The pneumatically-actuated RBO Hand 2 manipulating an abacus [158] (Copyright ? 2016, IEEE); (c) A universal gripper based on the jamming [159] (Copyright ? 2012, IEEE); (d)A gecko elastomer actuator gripper [160] (Copyright ? 2018, IEEE); (e) A 3-D-printed soft gripper with suction cup for effective grasping [161](Copyright ? 2019, IEEE).

The human hand can perform vigorous grasping because of its mechanism that combines the structure of a clamping component (multiple fingers) and a sustaining component(palm) [164]. Its mechanism and motion pattern have been well studied and considered to be an excellent example inspiring the design of robot grippers. There are many fabrication methods with different actuators adopted to permit similar morphology and thus replicate the similar motion pattern of a human hand with a robot gripper [158],[165]–[169]. Pneumatic actuators have been extensively applied in the design and fabrication of soft grippers for the shape adaptability and controllable stiffness, both desirable for dexterous and vigorous grasping [158], [165]–[167], [177].Deimelet al. [158] designed and fabricated a pneumatically actuated anthropomorphic soft robot gripper. With five fingers that can perform grasping posture and one pad serving as a supporting part, the presented prototype demonstrated the ability of dexterous and tight grasping of objects. It also exhibited good capacity of manipulation with objects with high weight, the performance of which was validated in multiple grasping tasks. Nagaseet al. [166] presented a similar design in their study that emphasized variable stiffness of robot fingers to overcome contradictory demand specifications regarding both dexterous manipulation and vigorous grasping of diverse objects. Ilievskiet al. [163]presented a gripper with a similar shape to a starfish, actuated by a Pneumatic Networks (PneuNet) based elastomer. The grasping-and-place behavior can be achieved by pressurizing the fingers to bend them inward and depressurizing them to recover their initial shapes. Nassouret al. [177] demonstrated an integral design of the soft gripper, where curvature and pressure sensors were mounted on pneumatically-actuated fingers to enhance interactive capabilities with the external environment and assess the shape information of both the gripper and object. Besides pneumatic actuators, smart materials also make a soft gripper functional. SMA-based actuators have been well studied. In [168], a soft robot hand that can imitate similar gripping performance of a human hand was proposed. The presented soft robot hand achieves bending performance by actuating embedded SMA strips sticking to the internal side of five silicone-made fingers. A similar design can be found in other studies. Wanget al. [170]presented a SMA-based robot hand with three fingers, each of which can achieve unidirectional bending performance. Two hinges on each finger can noticeably change stiffness, given the requirements of a grasping force in specific tasks. Nasabet al. [179] also demonstrated a method to change the stiffness,leveraging hybrid pneumatic actuator and elastomer strips.The presented gripper can be controlled with its grasping behavior of each finger being pneumatically actuated to perform bending. Lower and higher stiffness can be obtained by, respectively, heating up and cooling down the elastomer strips incorporated with the pneumatic actuator. Mantiet al.[167] adopted a simplified design principle in their study,where three paralleled accommodated fingers were unified for grasping tasks. Grasping behavior is realized when fingers simultaneously bend inward, with actively changed lengths of embedded cables. Bending performance can be achieved by driving one motor to control all three cable variables,employing a synergetic motion control strategy and open-loop control units, with no requirements for sensing. The fin ray effect refers to another well-known example in bionic design.This comes from the configuration and mechanism of a fish fin. Engineers have been inspired by this effect to design a flexible grasper, adapting to the shape of objects and performing vigorous grasps [180], [181]. A granular jamming based gripper is also well investigated this decade [159],[171]–[176]. These robot prototypes show versatile grasping ability of various objects with a simple control strategy,leveraging good shape adaptability inherent from their mechanism and good load capacity based on a vacuum process.

B. Modeling and Control Strategies for Soft Gripper

Rare studies have focused on theoretical dynamic modeling of soft grippers. The coordination motion pattern enhances coupling inside the mechanism, thereby increasing complexity of mathematical modeling work. A wide range of publications have presented open-loop and learning-based methods for grasping control [161,] [165], [169], [176], [182], [183](referring to Table III). It is argued that vigorous grasping can be realized by leveraging passive compliance even without an accurate contact model, instead of a traditional closed-loop control strategy with force feedback loop. Researchers have concentrated on upper-level control to plan a task-oriented motion sequence and provide accurate performance with given excitations [169], [182], while bottom motion control can be achieved by following learning-from-demonstration strategy or simple open-loop control strategy. Amendet al.[176] presented a gripper with two jamming actuators, each of which are treated as a robot finger, oppositely accommodated on the palm to clamp an object with combined action from vacuum suction. In their study, the authors analyzed the effective grasping force generated from friction and vacuum process although they gave no clues on how this could be utilized into the controller design. Tawket al. [161] also presented a teleoperated three-finger gripper unified with a suction pad to provide high load capacity. Force analysis was developed to model its payload in finger grasping and suction behavior. Fariaet al. [183] proposed a learning-based method for the presented pneumatically-actuated soft robot hand. In their study, the prototype was trained to imitate the demonstration of a human hand and replicate grasping performance. Similarly, Guptaet al. [165] enabled the soft hand to perform grasping through learning and imitating from human manipulation.

C. Discussion on Soft Gripper

Soft grippers provide good grasping performance with facile control strategies. A soft gripper can circumvent the planning of a grasping pose thanks to its shape adaptability that can easily ensure the force closure. Safe grasping manipulation is guaranteed without complex force control architecture due to its compliant mechanism. These merits enable increased investigation of this type of soft robot, especially when it comes to the manipulation of fragile objects. The morphology is essential to the performance of a soft gripper, while it demands joint contributions of advanced actuation methods ensuring rapid response and an optimized mechanism design providing vigorous grasping action. Compliance is the dominant property that enables desirable shape adaption and allows a soft gripper to have safe interactions without meticulous design of the controller, while at the same time decreasing the load capacity of the prototype. One should take into account this trade-off while considering specific demands of the task when integrating the material and corresponding mechanical design techniques. This issue is especially challenging when subtle manipulation is required, e.g., inhand manipulation. Better manipulation performance requires active shape and force regulation. This can be realized with embedded strain and tactile sensors. However, though numerous soft grippers have been proposed, few studies concentrate on in-hand manipulation. Commonly adopted underactuated mechanical systems additionally increase the complexity of manipulation controller design. Ongoing research concentrating on integration technology of material,mechanics and control would further enhance the application in real manipulation tasks in diverse environments.

IV. SOFT MOBILE ROBOT

Mobility and navigation in diverse terrains have long been studied to enhance the versatility of robots in multiple scenarios. Untethered soft robots operating in various environments have generated increasing interest in this decade. It is acknowledged that soft robots ensure prominent environmental adaptability due to their morphologies and bioinspired sensorimotor control patterns. Researchers have been inspired by living organisms that survive after long-term evolution and show transcendence in adaption to particular habitats. Multiple studies and reports can be found. Numerous soft robot prototypes that mimic the locomotion of their biological counterparts exist, exhibiting eminence when operating in specific working environments. The mechanical design of existing bio-inspired soft mobile robots demonstrates a strong correlation with the environment. Figs. 9–11 give several examples of the soft mobile robots.

A. Morphology

To be the fittest in natural selection, biological organisms continue to optimize their morphologies to specific habitats in generation-by-generation evolution and are eventually endowed with good morphology and sensorimotor capabilities. Researchers are inclined to fabricate a soft robot prototype with similar morphology to the living organisms so that the robot can reproduce their motion patterns and demonstrate distinct performance in specific conditions.Therefore, soft robots demonstrate various morphological characteristics, differentiated by their potential work environments. Subsequently, the paper will examine soft mobile robots by exemplifying typical types of commonly adopted morphology and motion strategies that enable them good mobility in target territories.

1) Underwater Bio-inspired Soft Mobile Robot

Fig. 9. Exemplified underwater soft mobile robot prototypes. (a) Octopuslike robot capable of swimming [184] (Copyright ? 2018, IEEE); (b)Leptocephali-inspired soft robot with dielectric elastomer actuators [185](Copyright ? 2018, IEEE); (c) Soft modular swimming robot with oscillating fins [186] (Copyright ? 2020, IEEE); (d) Soft underwater walking robot with active flow adaption by changing the shape [187] (Copyright ? 2019, IEEE).

Several studies have been primarily discussing on underwater bio-inspired robot design and locomotion strategies enabled by soft actuators [16], e.g., legged locomotion [184], [187]–[190], jet propulsion [191]–[193],undulating motion [185], [186], [194], etc. Marine living organisms have long inspired researchers to do mechanical design (as shown in Fig. 9) and study motion control method of underwater robots. The octopus, an example of marine invertebrate, features good performance in underwater environments. Known for its abilities to squeeze itself into a highly constrained environment, the octopus performs rapid and accurate reaching and fetching movement during its predation, and executes biped motion in the seabed. Thus,researchers have delved into the underlying principle of its motion achieved by its unique morphology, aiming to reproduce such advanced performance in a robot platform.Octopus-inspired soft robots that mimic the swimming and gait of their biological counterpart are studied. It is known that a robot exhibiting a similar shape to living organisms is more likely to intimate their locomotion. The bio-mimetic mechanical design of underwater soft robots inspired by the octopus has been presented in [184]. The prototype displays a spheroid shape of its major body with eight soft arms evenly distributed at its underside periphery. Other studies concentrating on octopus-inspired soft robots investigated the design method as well as locomotion strategies [188]–[190],[195]. The presented prototypes demonstrate both crawling[188], [189] and swimming [184], [190], [195] performance by learning motion patterns from its biological counterpart.On the whole, these motions are realized by coordination motion control of multiple soft arms rather than jet propulsion methods that were studied in [191], [192], [196]. It is well known that underwater environments impose extra disturbances on aquatic robotic systems. When a robot operates in such a situation, further study should be conducted to eliminate enhanced external interactions applied by the surrounding fluid. Otherwise, performance will be degraded,or the system can even collapse. In contrast, the specific underwater environment is conducive to some locomotion strategies. From the perspective of actuation, the underwater environment facilitates the operation of a specific type of actuator. Liuet al. [194] presented a design of a propulsion device in a similar shape to the fish fin. The locomotion performance of the proposed prototype was tested underwater,and its effectiveness in generating propelling force was shown. Meanwhile, the energy cost required for the generation of motion was minimized. Another instance is the ionic polymer metal composites (IPMC)-based actuation method, of which the working principle is ion motion with an applied external voltage differential. The external electrodes generate attractive and repulsive forces, thereby causing non-uniform distribution of ions based on the principle of “opposite attraction.” Subsequently, the deformation can then be achieved due to the difference in qualities of anthode and cathode. This actuator is capable of more effectively operating in aquatic environments, since moisture on electrodes in underwater environments can facilitate free motion of ions[16]. The aquatic environment is also known for enhancing agility by narrowing the cooling time for an actuator based on temperature-induced phase change, e.g., SMA. To this end,numerous prototypes have been developed leveraging these advanced-material actuators. With bio-inspired morphologies,several soft mobile robots are fabricated with similar motion patterns relative to their biology counterparts. It is acknowledged that IPMC- and SMA-based actuation methods lead to soft robotic mobility generated by in-turn undulation motion of fin-like or tail-like mechanisms [16]. The undulating or oscillating motion can also be generated by employing this actuation method. Unlike the motion strategies based on fin actuation, jet-propelled actuation, inspired by the motion pattern of sepia, is analyzed as well to perform a rapid movement in aquatic environments [191], [193].

2) On-the-ground Bio-inspired Soft Mobile Robot

In addition to robots working in aquatic environments,researchers have also been inspired by biological morphologies (e.g., multi-arms) that enable robust motion mode in rough terrains [197], [198]. Some researchers argue that the optimized performance of a soft robot cannot be achieved with artificial and heuristical mechanical and controller design. Instead, it should evolve its mechanism,motion pattern, as well as control algorithm with predefined target environments and specific tasks to eliminate “designers bias” [199], [200]. Corucciet al. [201] presented a selfevolutionary theory for a soft robot through simulation, to prove its feasibility in the design of morphology and locomotion strategy for a bio-inspired soft robot with multiple legs. Thus far, such a theory has not been reported in practical applications.

The study of motion patterns of the earthworm, caterpillar and other annelids has long been conducted. These organisms demonstrate agile and robust locomotion in various terrains,by leveraging their soft architectures to execute sequential contraction and extension motions of their body for forward,backward, and turning movement. Researchers, therefore, are inspired by the biomechanics as well as their locomotion strategies that are unified to obtain the impressive mobility in unstructured environments even if with complex terrains (as shown in Fig. 10). Several studies exist with the aim of replicating such performance in robot platforms [202]. A snake-like soft robot prototype was proposed in [203] and use slithering locomotion while leveraging friction of its contact surface.

Fig. 10. Exemplified on-the-ground bio-inspired soft mobile robot prototypes. (a) Fluid-driven soft robot permits inchworm crawling [197](Copyright ? 2020, IEEE); (b) An inchworm-inspired soft robot capable of omega-arching locomotion [204] (Copyright ? 2017, IEEE); (c) A legged soft robot capable of navigating unstructured terrain [198] (Copyright ? 2017,IEEE); (d) A soft robot with wall-climbing ability enabled by electroadhesive foot [205] (Copyright ? 2018, IEEE).

There is a universally adopted method of mechanical design of caterpillar-inspired and earthworm-inspired soft mobile robots. That is, the architecture of the robot displays the combination of two connection parts and one core body part.During the operation, deformation, either extension/contraction or bending, of the core body part enables stridelike motion with given periodic excitations based on the actively changing body length. In the meantime, one connection anchors to the environment, and the other moves with its core body part. This peristalsis-like forward or backward movement can then be achieved by regulating the action sequence that consists of the motions of three parts.The deformability of the core body part is critical to varying body length. In this light, researchers have studied the characteristics of different actuators to enhance the deformation effectiveness and agility. Soft actuators, namely SMA-based actuators [7], [206]–[210], electroactive polymer(EAP)-based actuators [9], [205], electromagnetic actuator[211], [212], and pneumatic actuators [204], [213]–[216] are implemented in soft robot systems which can then reproduce the motion pattern accordingly. Existing soft mobile robots following peristaltic motion patterns usually have similar morphology to that of an annelid. To achieve periodic bodylength variation of the robot, researchers combine elastic materials with smart soft actuation methods to mimic undulate or stretching behavior inspired by their biology counterparts.The function of the connection part that is to fix/disconnect the robot to the contact can be realized by actively enhancing/reducing the friction [204], [205], [208]–[210],[217]–[219]. According to an alternative strategy, the interchanging fixing and disconnection to the current contact can be achieved by a gripping and loosing action enabled by a unified gripper system [213]. The functions of these connecting and flex parts are unified in a feasible manner to permit forward, backward, and turning behaviors. They can then underpin applications in navigation and locomotion into multiple environments. A viable way to actively adjust the friction is to change the area of contact interface. Geet al.[217] presented the design and fabrication work of a pneumatically actuated soft robot. In their study, the fiber reinforcement accommodation was adopted to constrain the performance of the actuator to be unidirectionally stretched along the axial direction. Subsequently, the stride-based motion was then obtained by pressurizing or depressurizing its core body part, with joint contribution from connection parts.The anchoring to the contact of either the upfront or rear part was achieved by increasing the static friction in the contact interface, through pressurization-based expansion. Similar working principles can be found in a tube-climbing robot[220]. The presented robot prototype adopts the pneumatic actuation method as well. Accordingly, the performance of anchoring to the tube can be obtained by selectively pressurizing the top or bottom part. Alcaideet al. [219] also presented a soft robot that can perform both rectilinear climbing motion and turning behaviors. The robot consists of three cascade sections, each of which is made of silicone and is actuated with three embedded parallel-accommodated SMA strings. The radial expansion, to fix the robot inside the tube,and shape recovery, to obtain body length variable by reverting the core body from expanded to normal shape, can be achieved, respectively, by simultaneously heating and cooling all three SMA strings embedded in one section.Bending behavior can be achieved by partially actuating SMA strings in one section to change the orientation in the locomotion. Rozen-Levyet al. [213] demonstrated a unified design of the gripper system and core body. In the study, the contraction and shape recovery can be, respectively, achieved by tightening and loosing the embedded driven tendon. The three grippers, underneath each body section and inspired by the fin ray effect [180], [181], are shaped so that they can perform a robust grasping of various objects. Guet al. [205]presented a soft robot prototype that was capable of climbing upright walls. In their study, an EAP membrane was utilized to provide the length-change function. The frictions in the contacts to the wall can be actively controlled based on electrostatic forces, which is considered as a function of the applied external voltage. The velocity of the robot depends on the period of input signal sequence. The undulation motion pattern has been adopted in several studies [208], [217], [221],[222]. Wanget al. [221] presented a pneumatically-actuated earthworm-like soft robot. Bending performance was obtained through pressurization of the chambers linearly distributed along its core body part. The in-turn anchoring behavior of the front and backend of the robot was achieved by passive friction-changing ability. In their study, the front and backend sheet-like parts were designed to permit a certain angle relative to the baseboard to change the contact area in different motion phases. Such a method has been commonly employed in the mechanical design work of earthworminspired soft robots [193], [208], [217], [222].

3) Miniaturized and Micronized Soft Mobile Robot

Miniaturized and micronized soft robots have aroused increasing attention recently. Fig.11 demonstrates several instances of small-scale soft mobile robots. The mobility in highly constrained environments endows it with potentials in diverse applications [223]–[225], including drug delivery in medical use [226], [227], and subtle manipulation with cells in bioengineering use [228].

Fig. 11. Exemplified miniaturized and micronized soft mobile robot prototypes. (a) A magnet-driven origami robot capable of walking and swimming [229] (Copyright ? 2015, IEEE); (b) SMP-made four-limb soft robot with self-sensing ability [230] (Copyright ? 2020, IEEE); (c) Magnetically actuated microrobot with integrated tail and body morphology [231](Copyright ? 2016, Huang et al); (d) RUBIC [232] (Copyright ? 2019, Chen,et al).

Huet al. [227] demonstrated a millimeter-sized mobile robot capable of free and robust motion in diverse terrain even with the existence of obstacles. The robot is involved in simple architecture design in a sheet-like shape, characterized by silicone-made mechanism where magnetic particles are embedded. Therefore, reference motion and configuration can be obtained by regulating the magnetic magnet field.Miyashitaet al. [229] designed an origami robot and demonstrated its walking and swimming performance with applied magnetic fields. Made from shape memory polymer(SMP), the presented robot prototype was capable of thermally self-folding to permit shape adaption when executing different motion patterns. Liuet al. [230] also presented a SMP-made miniature soft robot with four limbs enabling crawling and jumping locomotion, for the sake of medical use. Huanget al. [231] have developed a magnetically actuated origami-inspired microrobot that permits self-folding, shape reconfiguration and motility, with an integrated tail and body morphology that enable better swimming efficiency. A mobile soft robot in a cubelike shape is presented in [232], which can realize rolling motion. In its design, each face of the presented robot consisted of four pneumatically actuated balloons. The locomotion is then generated based on flipping behavior when the balloons at one face are expanding. To set free the mobile robot from applied external (magnetic or electric) fields or torques, researchers have contributed to the design work of light-actuated robots[233], [234]. This type of untethered soft mobile robot characterizes sophisticated mobility with periodic light signals.

B. Modeling and Control Strategies for Soft Mobile Robot

Researchers tend to investigate characteristics of the actuators utilized in their prototypes and develop the mapping of system inputs and actuator performance. This work can be observed in previously reviewed publications in this chapter.It can be considered the first step to model the motion equation of the mobile soft robots. Due to a broad selection of types of actuators, there is no general modeling method since solutions vary with the working principles of adopted actuators and the geometry of mechanical structure. For instance, externally applied magnetic fields generate magnetic forces and torques along a specified direction and thus can effectively drive the magnet-actuated soft robot to move in different terrains and deform it to permit various locomotion tasks. Researchers should study how the external field results in material deformability with carefully selected magnetic fields [227] when controlling a magnetically actuated soft robot. Electromechanics should be studied in the EAP to model the strain caused by electrostatic effect. In this process,researchers should also pay attention to the viscoelasticity of nonlinear dielectric elastomer membrane between two electrodes. Conventionally, the stress attributed to electrostatic effect, can be modeled into the function of material permittivity and applied electric field. Thereafter, the contraction ratio of the membrane can be computed based on the constitutive law of the material [9]. Likewise,thermodynamics study is vital for actuators of SMA, SMP, or other thermally driven materials [235]. We will skip the detailed modeling method for each of these actuation methods. For more information, interested readers can refer to previously reviewed publications. The explicit relationship between system excitation and the corresponding robot motion in task space can be obtained by consolidating two models, which, respectively, depict how the actuator performs with applied excitations and how the robot acts in its task space with predictable actuator’s performance. Conventional modeling methods can be implemented in modeling robot motion with solved driving force generated from actuator performance. In other words, we can adopt the same modeling method for congeneric robots with a similar configuration,e.g., cephalopod-inspired or fish-like morphology in the aquatic environment [191], [193], [236], legged system with gait-based locomotion strategy [188], [189], cubelike shape capable of rolling motion [232], etc., as long as specific mapping of excitations and actuator performance is acquired.Indeed, we can refer to modeling methods in a rigid robot to address the common issue, which is to solve the motion equation of the robot system with given forces, torques, thrust,and other types of actuation. For instance, to model the motion equation of an underwater soft mobile robot given the propulsion forces generated by a fin-like actuator, researchers can also employ the energy-based modeling method [236].Rendaet al. [193] presented a shell-like underwater soft robot,in which the Cosserat rod theory was employed to model its motion equations. The major difference from rigid robot modeling methods is that the variance of internal energy caused by deformation should be considered in the case of soft robot. When mathematically modeling the robot systems that operate underwater or in other interactive environments,researchers should pay attention to sophisticated environmental interactions to more accurately explain robot locomotion. Hence, the integral model of a robot system can be obtained by unifying the upfront modeling processes,considering the underlying principle of the specific actuator adopted, and addressing common issues in modeling robot motion equations in the special working environment. The motion of a fish-like mobile robot, propelled by the undulation of IPMC-actuated fins, was mathematically modeled [236].The proposed model describes how the speed of the robot can be associated with the applied voltage if hydrodynamic effects act on the robot system. As for legged robots, whose motion is dependent on the reaction forces generated at the contact of the robot and the environments, a contact model with specific constraints is necessary to build motion equations and to guarantee preferred locomotion [187], [198]. In [237], the authors built the motion equation of the octopus-inspired robot and analyzed locomotion performance taking into account the propulsive force at contact as well as the hydrodynamic forces acting on the deformable mechanism. Simple open-loop control strategies are commonly adopted in locomotion of a soft mobile robot, with some exceptions in closed-loop studies to enable accurate motion control in the joint space [238],[239] and task space [240]–[242] provided that system dynamic/static models and sensing systems are available. A joint space closed-loop controller was proposed in [238] with embedded curvature sensors providing state feedback to enable the rolling motion of a circular-shape robot, yet this could hardly guarantee the error convergence in the task space. Several studies concentrate on the path following of miniature soft mobile robots actuated by imposed external(electrical or magnetic) fields [240]–[242]; thus one can enable desired locomotion in the task space. A visual servoing scheme proved viable in path-following tasks [241], where a stereo vision system provided real-time 3D position feedback of a millimeter-scale soft film robot and the proposed controller could enable the position error convergence in the task space.

TABLE IV SOFT MOBILE ROBOTS

C. Discussion on Soft Mobile Robot

Despite the progress in the design and fabrication of soft mobile robot, there is no report on a general mathematical modeling method for soft mobile robots as seen in Table IV.The possible reason of this may be attributed to various actuators adopted in robot systems. The mapping between the driven forces applied on the robot and real excitations, e.g.,external field, propulsion forces from body undulation/oscillation, should be specifically determined. Diverse morphologies (contradictory to a soft continuum robot usually enjoying the same characteristics in the configuration) make it impossible to construct the mapping of configuration space and the Cartesian space using an identical modeling method.The motion can be solved with an exactly known driving force considering the unique structure of mechanism, by adopting mathematical modeling methods leveraging equilibrium of forces/momentum/energy, or FEM method, etc.

Interestingly, soft mobile robots overlap with soft continuum robots in a small part, e.g., a legged mobile robot actuated by soft actuators. Merging with deformable soft legs,this type of soft robot can better adapt to different terrains and environments by actively shape their configuration, and thus can provide robust walking performance. The configuration of the individual leg can be depicted by following the same method with that of a soft continuum manipulator. However,when modeling the motion of a legged mobile robot, emphasis should be put on the coordination of multiple legs. Different working environments faced them also hinder the general modeling methods. Sometimes environmental interactions should be taken into account if it exerts dominant effects on robot motion. For instance, fluid dynamics should be considered when it comes to the underwater working environment; friction and contact model resulting from environmental interaction should be solved in modeling and controlling legged soft robots. In summary, the solution of mapping in between actuation and configuration spaces comes from the analysis of material characteristics under external/internal excitations; the mapping between configuration and task spaces depends on the geometry of the mechanism as well as specific environmental effects.Therefore, when it comes to the problem of modeling and control of a soft mobile robot, viable methods are highly dependent on the multifactor functioning from their material,mechanism and predefined working conditions.

V. CONCLUSION AND DISCUSSION

Soft robotics has shown its continual development in this decade. Thus far, the unfaded impetus has driven the field of robotics into a new era with increased intelligence and humanization. On-going research is elevating the level of techniques from various disciplines. The range of soft robotics is too broad to be concluded sufficiently here. This paper places emphasis on a very small part of it, as an attempt to more intuitively guide interested researchers to quickly learn about the current progress in the mainstream of soft robot prototypes and the way of designing a soft robot system to implement practically.

Inspired by the morphology of living organisms, soft robots are more likely to replicate motion patterns and satisfy the demands of control tasks in various terrains. Their performance depends on many factors. Evaluating metrics thus should cover aspects of design (flexibility, load capacity,repeatability, response rate, etc.) [243], modeling(fidelity,real-time computing ability, etc.) [89], motion control(accuracy, converging rate, robustness, etc.) [244], and manipulation (various objects grasping, grasping force, etc.)[245]. Researchers are supposed to solve common issues,which include modeling accuracy, interactive ability, real-time controllability, etc. The emphasis should also be placed on nontrivial issues attributed to specific characteristics in the compliant mechanism. From the perspective of system modeling, these issues cover high nonlinearity of materials,hysteresis in electromechanics and thermodynamics, strong coupling effects that ubiquitously exist in the nonlinear systems, etc. Though numerous research has been conducted on modeling hysteresis mathematically with known material parameters, the tedious and costly offline calibration work for nonlinear elastic materials is required and will complicate applications of a soft robot if a model-based controller is employed. Conventional offline methods sometimes suffer from calibration errors, which degrades modeling accuracy.Control methods for soft robots have been studied these years,e.g., model-based control algorithms [34], [131], model-less control based on optimization [129] or online estimation [32],and learning-based open-loop control [69], [72]. They do exhibit satisfying performance in specific environments but have their own limits. For instance, researchers usually simplify the system modeling by prescribing some assumptions, e.g., small perturbations to simplify system nonlinearity, low-speed motion to disentangle coupling of system model, etc. Given real-time requirements, the complexity of the hyper-redundant system model should be reduced at the cost of the performance degradation. Limited adaptability of a control algorithm to various environments is another issue that impedes further enhancement in the applicability of soft robots. A possible solution originates from the development of robust control algorithms combined with more advanced sensing methods that improve the ability of environmental cognition. Actuators based on smart materials sometimes show delays in actuation. Therefore,further study on advanced materials is also required to improve the performance of actuators.

Soft robotics is considered to be a highly interdisciplinary subject; breakthroughs thus require closer multi-discipline collaboration. Most existing publications have studied the design and fabrication of soft robot prototypes from bioinspired approaches. Due to the internal relationship between the morphology and sensorimotor control, the presented robots have the potentials to perform similar motions to those of living organisms. By gaining insights into the underlying principle of morphology, locomotion, and sensorimotor control, the replicated motion patterns are more likely to be achieved on soft robot platforms. They can then demonstrate intelligence, agility, robustness on par with those of their biology counterparts. More functional materials lay the basis for the development of soft robot prototypes. Optimized mechanical design helps to better learn bio-inspired motion strategies. Thereafter, having a control methodology enhances their performance in accuracy and versatility, with computer technology jointly contributing to improving the capabilities of real-time planning, sensing, recognizing and acting.

This paper presents a review work on the scientific studies of soft robotics, in the framework of commonly adopted morphologies and corresponding motion strategies, aiming to help new researchers easily step into this domain.Nevertheless, there is still a long way ahead for prospective applications. This can be achieved with more efforts and more cooperation devoted by researchers from diverse backgrounds.