2.1.1. Types of soft robots

Table I summarizes the common types of soft robot types and their acronyms to be used in the rest of the text. Figure 2 shows some examples of soft robots found in the literature, including bio-inspired robots [Reference Schiller, Seibel and Schlattmann23, Reference Youssef, Soliman, Saleh, Elsayed and Radwan24], modular robots [Reference Medvet, Bartoli, De Lorenzo and Seriani25, 26], grippers [Reference Phanomchoeng, Pitchayawetwongsa, Boonchumanee, Lin and Chancharoen27, Reference Chen, Li, Yan, Xie, Feng, Zhu and Zhao28], and continuum robots [Reference Costa and Reis29, Reference Van Pho, Dhyan, Mai, Han and Chow30]. Note that these categories are not mutually exclusive: some soft robots can combine these categories or their technologies to achieve specific functionalities according to their task.

Table I. List of acronyms for type of robot categories.

Figure 2. Some examples of common soft robot types found in the literature: bio-inspired (a) gecko-like soft robot [Reference Schiller, Seibel and Schlattmann23] and (b) Soft swimming fish [Reference Youssef, Soliman, Saleh, Elsayed and Radwan24]; modular soft robots (c) [Reference Medvet, Bartoli, De Lorenzo and Seriani25] and (d) [26]; soft grippers (e) [Reference Phanomchoeng, Pitchayawetwongsa, Boonchumanee, Lin and Chancharoen27] and (f) [Reference Chen, Li, Yan, Xie, Feng, Zhu and Zhao28]; and continuum soft robots (g) [Reference Costa and Reis29] and (h) [Reference Van Pho, Dhyan, Mai, Han and Chow30]. All images were used under CC-BY license.

Bio-inspired soft robots mimic the structure, motion, functionality, or behavior of living mechanisms, such as fish [Reference Abbaszadeh, Leidhold and Hoerner31], snakes [Reference Tesch, Schneider and Choset32], worms [Reference Tang, Huang and Li33], quadrupeds [Reference Pollayil, Della, Mesesan, Englsberger, Seidel, Garabini, Ott, Bicchi and Albu-Schaffer34], lizards [Reference Schiller, Seibel and Schlattmann35], and insects [Reference Ahn, Liang and Cai36]. They can exhibit a variety of locomotion styles found in nature, such as swimming, crawling, flying, or slithering – depending on the living creature they are mimicking. This allows the robots to navigate through complex environments with great agility and adaptability. Their drawback is the complicated control schemes they require to achieve specific movements precisely. Additionally, some methods might be energetically demanding to accomplish bio-inspired locomotion, which can be improved through optimization techniques

Cable/tendon-driven soft robots use tension cables as their main actuation mechanism [Reference Bern and Rus37, Reference Ros-Freixedes, Gao, Liu, Shen and Yang38], pulling the robot body to bend, stretch, and deform. They are usually lightweight, compact, compliant, and capable of safe interaction. They can be particularly useful for creating fine and precise movements in applications such as medical operations – thanks to their distributed actuation. Their drawbacks include the need to replace and maintain the cables as they wear out with time and usage. Additionally, controlling the joint movements is more difficult compared to traditional rigid robots due to the distributed cable system throughout the actuated section of the robot body.

Fluid-driven soft robots use fluidic pressure, such as pneumatic or hydraulic pressure, for actuation [Reference Do, Banashek and Okamura39, Reference Marchese and Rus40]. The actuators fill up with the fluid medium to change the shape of the soft robot and achieve motion and manipulation. Due to the fluids’ inherent properties, they are highly compliant and can achieve safe and soft interaction with the environment. Moreover, they can adapt well to unstructured and dynamic environments. Their drawback is mostly the complexity of the control system due to the nonlinear behavior of fluid dynamics. Additionally, despite their capability of achieving quick movements, the speed of fluid-based soft robots may not always match that of rigid-link robots, especially in applications that require high-speed precision.

Soft grippers are an adaptable class of end effectors with flexible and soft materials that can grasp objects of different sizes and shapes. They usually consist of soft fingers [Reference Li, Chen, Song, Liang, Zhu and Chen41], soft-granular material [Reference Fitzgerald, Delaney, Howard and Maire20], or magnetic-based inflatable systems [Reference Stroppa, Selvaggio, Agharese, Luo, Blumenschein, Hawkes and Okamura42]. Their advantages include manipulation with a gentle, humanlike touch, robustness, and compliance. Depending on their actuation mechanism, the complexity of the control scheme can vary. One of their drawbacks is having a limited payload – achieving the necessary force for grasping different objects can be challenging. Additionally, complex designs might be required that can adapt to grasping a wide range of various objects, and additional sensors might be needed.

Soft manipulators, as opposed to conventional rigid manipulators, are designed with soft materials – such as fabric [Reference Stroppa, Selvaggio, Agharese, Luo, Blumenschein, Hawkes and Okamura42, Reference Coad, Blumenschein, Cutler, Zepeda, Naclerio, El-Hussieny, Mehmood, Ryu, Hawkes and Okamura43], plastic [Reference Do, Banashek and Okamura39], elastomers [Reference Marchese and Rus40, Reference Thongking, Wiranata, Minaminosono, Mao and Maeda44], or flexible polymers [Reference Akcan, Erdil, Korkut, Baran and Gokdel45] to produce compliant movements.

Continuum soft robots are manipulators that resemble the movements and structures of organisms found in nature, such as octopus tentacles [Reference Laschi, Cianchetti, Mazzolai, Margheri, Follador and Dario46] or elephant trunks [Reference Atia, Mohammad, Gameros, Axinte and Wright47]. Their continuous bodies can perform smooth and continuous movements, such as bending and moving seamlessly without discrete segments. They are typically made up of elastic and flexible segments connected. They offer distributed actuation, similar to fluid and cable-driven soft robots, and are capable of performing multi-DoF motions. Continuum robots can be considered as manipulators. The drawback of these devices is the complexity of the control and modeling problem imposed by their continuum nature.

Material-property-actuated soft robots are actuated by altering the properties of the materials they are composed of, which are responsive to an external stimulus. The materials are selected to respond to a specific type of external stimuli, thus actuating the robot in the desired fashion. There is a wide variety of materials that can be used to manufacture this category of soft robots, including piezoelectric, magnetically responsive, heat sensitive, light sensitive, and chemical materials. The selection of a suitable material can be done according to the application of the robot and the way the robot is intended to move. For instance,

• • Piezoelectric materials induce controlled mechanical deformations as a response to electrical currents [Reference Atia, Mohammad, Gameros, Axinte and Wright47] to provide precise and rapid reversible movements. They can either be embedded into the entire structure of the robot or fixed to specific regions. They are great at offering precise and rapid actuation in complex movements, thus quick and controlled movements. They are generally small in size and lightweight and can be used in soft robots without limiting flexibility. However, most of these materials are prone to mechanical fatigue and thus permanent deformation after being exposed to high strain magnitudes.

• • Magnetically responsive materials are embedded or coated on soft materials, which respond to the magnetic field and produce various types of movements (i.e., bending, twisting, or extending) [Reference Tang, Huang and Li33, Reference Shin, Choi, Jeon, Choi, Song and Park48]. The type of movement is determined by the direction, pattern, and intensity of the magnetic particles and field. Their soft nature and precise motion ability are extremely useful for delicate tasks in confined and complex environments (e.g., for surgical applications). Moreover, they do not require additional external power cables, rendering the overall system lightweight and small in size. However, they feature the drawback of a complex control algorithm with precise calibration and a limited speed. Additionally, the response of the magnetic particles may be heterogeneous due to the material structure, and maintaining a uniform response across the robot’s structure may be challenging.

Modular soft robots are composed of individual modules that can be interconnected and reconfigured to create diverse robots with different functionalities [Reference Pigozzi, Tang, Medvet and Ha49, Reference Kimura, Niiyama and Kuniyoshi50]. Each module is independent from the other modules and self-contained with an independent control unit, power cell, sensors, and actuation mechanism. When these independent modules are combined, the modular soft robots can adapt to different shapes and environments. They provide flexibility in design that creates novel structures to suit the task, scalability (i.e., the robot might be scaled up or down by adding or removing modules), increased robustness due to the increased redundancy (i.e., such that a malfunctioning module does not result in a complete breakdown of the robot), and ease of maintenance (i.e., a problematic module can be replaced individually without the need to disassemble the entire robot). Their drawbacks include an increased complexity in the control scheme to achieve seamless coordination between the modules and the need for standardization of communication and control protocols among the modules.

2.1.2. Soft robot applications

The applications of soft robots can extend to a variety of different fields. We classified them into three main categories as summarized in Table II with their acronyms that will be used in the rest of the text.

Table II. List of acronyms for applications of soft robots.

Exploration/locomotion/tracking applications take advantage of soft robots thanks to their adaptability and soft/compliant nature. They are widely used for the exploration of unknown terrains and cluttered environments such as narrow spaces [Reference Coad, Blumenschein, Cutler, Zepeda, Naclerio, El-Hussieny, Mehmood, Ryu, Hawkes and Okamura43] or search and rescue operations in dangerous and/or debris-filled regions that traditional rigid robots may find challenging to navigate [Reference Ng, Tan, Xu, Yang, Lee and Lum51].

Grasping and object manipulation applications are well-suited for soft robots since they can deform to grasp and move objects or manipulate the environment safely due to their flexible and soft nature. These properties allow them to bend and accommodate the shape of the object to be grasped and thus moved easily. This can effectively be performed by soft robots on delicate and soft objects [Reference Stroppa, Selvaggio, Agharese, Luo, Blumenschein, Hawkes and Okamura42, Reference Stroppa, Luo, Yoshida, Coad, Blumenschein and Okamura52], for multi-object grasping [Reference Choi, Schwarting, DelPreto and Rus53], or for picking up samples from hazardous environments [Reference Coad, Thomasson, Blumenschein, Usevitch, Hawkes and Okamura54].

Medical surgery and procedure applications can take advantage of soft robots due to their soft structures [Reference Gerboni, Greer, Laeseke, Hwang and Okamura55, Reference Talas, Baydere, Altinsoy, Tutcu and Samur56]. In these applications, soft robots travel through and manipulate different parts of the human body without causing harm and injury to the internal organs, unlike rigid robots. It is worth mentioning that soft robots designed for medical surgery and procedure applications can be perceived as a subfield of grasping and manipulation robots with a special aim.

2.1.3. Optimization metrics

Table III summarizes the classification of the metrics and aspects of soft-robot designs that are optimized in the literature with acronyms that will be used in the rest of the text.

Table III. List of acronyms for optimization metrics.

Topology of the design is directly altered to yield the optimal shape of the robot through a comprehensive exploration of the design possibilities in terms of structural configurations, material selection, and transducers to maximize the overall performance of a soft robot. This can include the geometry, configuration of the robot, morphology, and center of gravity. To facilitate this, designers can employ advanced computational design and simulation tools for analyzing and predicting the responses of different topological configurations. Optimizing the topology is key to expanding the capabilities of a soft robot and intelligently making use of its flexibility.

Kinematics of the design refers to the relationship between the angles of deformable joints, linkages, and geometric configurations, which contribute to understanding the range of motion (i.e., the workspace of the robot) and maximizing it. Obtaining the kinematics of a robot often requires complex modeling of the geometric structure and the segmentation of the robot, along with simulations, as well as optimization methods to understand and predict the complex interactions among the joints, actuators, and flexible structures. A well-defined kinematics relationship ensures whether a specific motion is achievable as well as finding the most efficient way to achieve it. However, when the robots feature soft components or innovative DoFs such as growth and retraction, the kinematics become more complex and different than the ones of rigid robots [Reference Stroppa57].

Output forces of the soft robot design can be maximized (e.g., grasping force in the case of grippers) through optimization methods. While optimizing the forces for the soft robots, achieving a certain output force is challenging because many factors need to be taken into account, such as the nonlinear nature of soft materials, the selection of actuation methods to obtain the desired force, the geometry of the soft structure, the control architecture, and many other aspects to attain and amplify the available forces.

Locomotion of the soft robot design can be optimized to achieve a correct balance among material properties, control schemes, and actuation mechanisms. To obtain efficient and versatile movements, it is crucial to overcome the challenges imposed by the soft materials and actuation methods – such as actuation type, speed, locomotion gaits, control type, and power management [Reference Sun, Abudula, Yang, Chiang, Wan, Ozel, Hall, Skorina, Luo and Onal58]. Designers must exploit the nature of the flexible materials to make robots adaptable to different terrains, such that future research will expand the applications in this area.

Sensors of the soft robot design can be optimized in terms of placement, type, and number. This optimization is crucial for maximizing the adaptability and performance of soft robots. This is due to obtaining more useful data from sensors to be used in purposes such as control feedback [Reference Ferigo, Medvet and Iacca19] as well as kinematics and shape estimation in real time while eliminating the need for complex models of soft robots [Reference Kim, Ha, Park and Dupont59].

Energy optimization in soft robot design is essential for enhancing the efficiency and sustainability of the systems to be designed. That is, the total energy consumption can be minimized during deformations of soft robots to obtain efficient and precise movements in various tasks. Moreover, considering the energy consumption during the design stage can provide soft robotic systems with an extended operational life.

2.2.1. Optimization problems

Optimization is the process of solving a problem expressed as in Eq. (1). Specifically, it consists of finding a particular set of decision variables ( $x$ ) that meet specific constraints (inequalities $g_j$ , equalities $h_k$ , and bounds) while minimizing or maximizing one or more objective functions ( $f_m$ ).

(1) \begin{align} & \mathrm{minimize/maximize} \quad \{f_1(x),{\ldots },\ f_M(x)\} \nonumber \\ & \quad\qquad\qquad\qquad\qquad x = \{x_1,{\ldots },x_N\}^T \nonumber \\ & \quad\qquad\qquad\qquad\qquad x_i \in x, \quad i=1,{\ldots },N \nonumber \\ & \qquad\qquad \mathrm{subject\ to}\quad g_j(x) \geq 0, \quad j=1,{\ldots },J \nonumber \\ & \quad\qquad\qquad\qquad\qquad h_k(x) = 0, \quad k=1,{\ldots },K \nonumber \\ & \quad\qquad\qquad\qquad\qquad{x_i}^L \leq x_i \leq{x_i}^U \end{align}

When more than one objective function is involved in the optimization problem, we define it as a multi-objective optimization problem (MOOP). In most cases, all these objectives conflict with each other, meaning that the optimization of one objective happens to the detriment of another one. This makes it difficult to optimize one objective without worsening others. Therefore, instead of obtaining a single optimal solution, the MOOPs require the retrieval of trade-off solutions, which are referred to as the Pareto optimal front [Reference Deb60, Reference Miettinen61]. These solutions that lie on the Pareto optimal front are non-dominated by other solutions – meaning no other solution is better in all objectives. On the other hand, trade-off solutions might be optimal for one objective but not for others [Reference Kalyanmoy62, Reference Hartikainen, Miettinen and Wiecek63].

2.2.2. Optimization methods: exact versus approximate methods in engineering

Algorithms or procedures solving optimization problems, namely, optimization methods, can be mainly classified as exact or approximate methods. Exact methods retrieve the exact optimal solution to a problem [Reference Deb64], whereas approximate methods obtain suboptimal solutions that are approximations of the exact optimal solution. Engineers favor the latter to solve design problems due to the complexity of the objective functions. Designing robotic devices – such as soft robots – is a complex task that usually involves a considerable amount of variables and parameters. This makes the effectiveness of exact methods not always guaranteed [Reference Norde, Patrone and Tijs65], especially considering that these methods are gradient-based and require the objective function to be differentiable [Reference Bonnans, Gilbert, Lemaréchal and Sagastizábal66] – which is mostly not the case in real engineering problems. Furthermore, reliable and robust solutions are preferred over exact ones: slight changes in the value of the optimal solution may result in overall instability [Reference Nomaguchi, Kawakami, Fujita, Kishita, Hara and Uwasu67], and uncertainty in the search space may lead to unfeasible optimal solutions [Reference Dizangian and Ghasemi68].

While exact methods are usually based on math, most of the approximate methods are based on heuristic search [Reference Edelkamp and Schrödl69]. A heuristic is an estimation of how close a point in the search space is to the optimum, and it can be expressed in many forms such as geometrical distance in the space or state evaluation functions. While many approximate methods are heuristic-based, under certain conditions, heuristic methods can guarantee optimality and therefore be considered exact – for example, the path planning algorithm A^* [Reference Hart, Nilsson and Raphael70] can find the optimal path if its heuristic is admissible and consistent [Reference Russell and Norvig71].

2.2.3. Evolutionary computation

EC is a branch of artificial intelligence focusing on solving optimization problems based on some of the most popular approximate techniques in engineering [Reference Stroppa, Soylemez, Yuksel, Akbas and Sarac22, Reference Kalyanmoy62] – Figure 3 shows a partial taxonomy of EC methods. EC is mainly inspired by Darwin’s theory of natural selection, in which individuals adapt to their habitat to ensure the survival of the species – therefore, EC techniques are considered as metaheuristic algorithms. EC techniques mimic this process by randomly generating a population of potential solutions (individuals) to an optimization problem and then gradually “evolving” them toward the optimal one. In algorithmic terms, individual evolution simply consists of mixing and changing numerical values to decision variables until better solutions (i.e., with higher/lower value of the objective function) are found. The specific process depends on the algorithm’s specifics, and EC encompasses a wide range of algorithms [Reference Back and Schwefel18, Reference Dumitrescu, Lazzerini, Jain and Dumitrescu72, Reference Fogel73]. The flowchart depicted in Figure 4 shows the generalized framework of any EC algorithm (applicable to evolutionary algorithms, swarm intelligence, etc.).

Figure 3. Evolutionary computation taxonomy with some of the most relevant algorithms.

Figure 4. Generalized flowchart framework of any algorithm in evolutionary computation.

Because EC techniques propagate several solutions concurrently rather than just one, they fall under the population-based methods [Reference Wu, Mallipeddi and Suganthan74]. This property allows algorithms to obtain several optimal solutions when needed, which applies to multimodal (i.e., when also local optima are sought in addition to the global one) or conflicting MOOPs (i.e., when more than one objective is involved). This property also allows EC to parallelize the process and cover a larger area of the search space while avoiding premature convergence [Reference Deb75].

Lastly, engineers are also drawn to EC techniques because they simplify the formalization of problems. These techniques can be applied to non-differentiable functions, work without the need for precise mathematical equations in their objectives, and are effective regardless of the mathematical formulation of the problem. This makes the methods independent from the specific optimization problem, such that it is possible to utilize the same technique in diverse scenarios. This is particularly helpful in mechanical design, where the design process might be dependent on a model simulation [Reference Baris Akbas, Soylemez, Zyada, Sarac and Stroppa76].

3.1.1. Evolutionary algorithms (EAs) and genetic algorithms (GAs)

EAs, particularly GAs [Reference Goldberg77, Reference David78], rank among the most popular EC techniques. They mimic the natural process of selection and the survival of the fittest [Reference David78]. These algorithms (i) create a group of random solutions within the problem’s search space, referred to as a population of individuals or chromosomes to resemble genetic processes, (ii) assign a fitness value to each solution by evaluating the objective function, and (iii) generate new solutions by combining the values of individuals from the current population, a process known as crossover. By allowing only the fittest individuals to undergo crossover and be retained in future generations (i.e., the following iterations of the algorithms), GAs direct their population toward the optimal solution (i.e., evolve). The crossover operation exploits the strengths of good solutions (i.e., the most fitting), facilitating fasting convergence to a (sub)optimal solution. However, this process does not guarantee the retrieval of the global optimum and may get stuck to a local one. To address this issue, GAs incorporate a genetic mutation operator, inspired by natural mutation, which randomly alters the values of a newly generated solution. This promotes exploration of the search space and allows the algorithm to escape from local optima. Additionally, some GA implementations retain the most fitting individuals in the population, such that they can be used in the following generation for crossover to exploit promising regions of the search space (a strategy known as elitisim); other implementations do not allow any individual of the previous generation to be retained in the new one – that is, the new generation is only composed of the offspring generated by crossover and mutation – allowing for a more extensive exploration of the search space albeit storing the most fitting individual separately from the population (a non-elitist strategy).

Apart from the benefits of population-based techniques (see Section 2.2.3), GAs offer the advantages of being simple to implement and highly effective in converging toward (sub)optimal solutions. The biggest disadvantage is that their many genetic operators come with many parameters, and fine-tuning these parameters can be challenging – often done by trial and error. Furthermore, due to their iterative stochastic nature, GAs are not an efficient choice for real-time optimization.

3.1.2. Swarm intelligence (SI) and particle swarm optimization (PSO)

SI is an EC technique inspired by swarms, herds, or flocks of animals gathering to locate food or build colonies [Reference Yang122]. The population of individuals – in this case, the swarm – features a collective behavior and follows the trend of the most fitting individual, exploiting its information for convergence. The most used method in SI is PSO [Reference Kennedy and Eberhart82], which mimics a flock of birds changing their direction based on the one locating food, forming a choreography of individuals pointing at the optimal solution. Each individual in PSO changes its search pattern by learning from the rest of the swarm’s behavior. This cooperative and iterative process makes the swarm converge to the optimal solution. When compared to EAs, PSO is computationally inexpensive in both time and space, and it often shows better performance, a higher convergence rate, and higher-quality solutions [Reference Kumar and Gupta123, Reference Ou and Lin124]. PSO also features different multi-objective optimization versions [Reference Coello119], which however suffer from poor convergence to the Pareto front and with insufficient diversity.

In our literature review, we found the following research studies that used PSO to optimize the design of soft robots:

• • Abbaszadeh et al. [Reference Abbaszadeh, Leidhold and Hoerner31] used PSO to optimize the strain gauge of a soft-aquatic robot for kinematics function regression;

• • Chikhaoui et al. [Reference Chikhaoui, Granna, Starke and Burgner-Kahrs108] used PSO to optimize the curvature of two robotic arms by maximizing the number of collaborative configurations among the ones retrieved with forward kinematics;

• • Djeffal et al. [Reference Djeffal, Amouri and Mahfoudi109] and Merrad et al. [Reference Merrad, Amouri, Cherfia and Djeffal110] used PSO to optimize the accuracy of the kinematic computation for the continuum robot’s design by minimizing the displacement between the end effector and the target;

• • Atia et al. [Reference Atia, Mohammad, Gameros, Axinte and Wright47] used PSO to optimize the design of an elephant-trunk-inspired soft robot by minimizing the differences between the desired robot shape and the current one;

• • Chen et al. [Reference Chen, Ma, Zhang and Liu111] used PSO to optimize the path planning and trajectory tracking through inverse kinematics of a modular manipulator (could be either soft or rigid) by minimizing its maximum joint bending angles at any moment; and

• • Gao et al. [Reference Gao, Hao, Yang, Cheng and Xiang112] used PSO to optimize the accuracy of the kinematic computation for a hybrid manipulator by minimizing the displacement between the end effector and the target.

3.1.3. Differential evolution (DE)

DE is used for problems having continuous domains. Individuals are represented as vectors, and perturbations are applied with vector arithmetics to diversify the population and explore the search space [Reference Storn and Price83]. However, when dealing with MOOPs, studies have shown that DE is less efficient than other MOEAs as it does not converge to the Pareto optimal front but to a nearby suboptimal and non-diverse front.

In our literature review, we found the following research study that used DE to optimize the design of soft robots:

• • Tan et al. [Reference Tan, Gu and Ren114] used DE to calibrate a soft robot hand, specifically for the kinematics of the finger and the relationship between fingers. They compared DE with other two non-EC techniques: IPA (see Section 3.2.1) and LMA (see Section 3.2.2), claiming that these two methods outperformed DE in terms of identification accuracy and processing time.

3.1.4. Estimation of distribution algorithm (EDA)

The EDA [Reference Larrañaga and Lozano84] is an EC technique exploiting probabilistic models in the search process. Unlike traditional EC methods that rely on crossover and mutation to create offspring, EDAs rely on the concept of building and refining probability distribution that characterizes the relationships between variables in the search space [Reference Pelikan, Goldberg and Cantú-Paz125]. This distribution is then used to generate candidate solutions – that is, the distribution of high-performing solutions guides the exploration of the search space, in a sort of knowledge transfer within individuals in the population. Thanks to this process, EDA can efficiently locate optimal solutions and converge with few iterations.

In our literature review, we found the following research study that used EDA to optimize the design of soft robots:

• • Cheong et al. [Reference Cheong, Ebrahimi and Duggan113] used EDA to optimize the design and kinematic calculation of a multi-joint continuum robot by minimizing the total actuator torque applied at the base of each continuum joint.

3.1.5. Evolution strategy (ES) and covariance matrix adaptation evolution strategy (CMA-ES)

ESs [Reference Rechenberg126, Reference Hansen, Arnold and Auger127] were introduced specifically for design optimization problems [Reference Rechenberg126, Reference Lichtfuss128, Reference Schwefel129]. Their working principle is similar to that of a GA without the crossover operator, where new solutions are sampled based on a multivariate normal distribution, and pairwise dependencies between variables are represented by a covariance matrix. The CMA-ES [Reference Hansen85, Reference Hansen and Ostermeier86] is a method to update ES’s covariance matrix, which allows the algorithm to dynamically adjust the exploration and exploitation of the search space.

In our literature review, we found the following research studies that used CMA-ES to optimize the design of soft robots:

• • Medvet et al. [Reference Medvet, Bartoli, Pigozzi and Rochelli90] used CMA-ES to optimize the morphology of a voxel-based soft robot by maximizing the distance it traveled during a locomotion task;

• • Ferigo et al. [Reference Ferigo, Iacca and Medvet116] used CMA-ES to optimize the sensory apparatus (i.e., the type of sensors and number) of a voxel-based soft robot by maximizing its average speed during a locomotion task; and

• • the same authors [Reference Ferigo, Medvet and Iacca19] published a different study using CMA-ES to optimize the location of sensors in a voxel-based soft robot by maximizing the intensity of their babbling (i.e., the phase of the robot’s life when it explores its sensing ability).

3.1.6. Adaptive stochastic search (ASS)

ASS [Reference Zhou and Hu93] is a method for solving non-differentiable optimization problems by converting them into a differentiable one on the parameter space of the parameterized sampling distribution; then, it uses a direct gradient search method to find fitter solutions – that is, it approximates the gradient of the objective function by evaluating random perturbations around some nominal value of the variable of optimization. Although this method does not directly fall within the area of EC, we categorized it as such due to its many similarities with EC (e.g., being population-based).

In our literature review, we found the following research study that used ASS to optimize the design of soft robots:

• • Exarchos et al. [Reference Exarchos, Wang, Do, Stroppa, Coad, Okamura and Liu118] used ASS to optimize the design of a soft growing robot (i.e., solve the inverse kinematics) for a specific task by minimizing the error between the end effector and desired target, in tasks having multiple targets and obstacles.

3.1.7. NeuroEvolution

NeuroEvolution refers to strategies based on machine learning and neural networks [Reference Abdi, Valentin and Edelman130] in which the weights and topology of the network are evolved by EAs.

3.2.1. Interior-point algorithm (IPA)

IPAs [Reference Byrd, Hribar and Nocedal133, Reference Wright151] are mostly used for convex optimization problems [Reference Byrd, Hribar and Nocedal133]. They are iterative methods that retrieve the optimal solution by traversing the interior of the feasible search space (i.e., within the problem’s constraints). Remarkably, IPAs can solve linear programming problems in polynomial time [Reference Wright151]. However, their gradient-based nature requires the objective function to be differentiable – implementations for non-differentiable functions can also be found in the literature [Reference Kischka, Lorenz, Derigs, Domschke, Kleinschmidt, Möhring, Goffin and Vial173]). There are even readily available software packages for nonexpert users of these methods: for example, MATLAB Optimization Toolbox offers an implementation of IPA with the commands fmincon, GlobalSearch, and MultiStart.

In our literature review, we found the following research studies that used IPA to optimize the design of soft robots:

• • Tan et al. [Reference Tan, Gu and Ren114] used IPA in a study for the calibration of a soft robot hand, specifically for the kinematics of the finger and the relationship between fingers. They compared IPA with other two techniques: LMA (see Section 3.2.2) and DE (see Section 3.1.3), claiming that both IPA and LMA outperformed DE – the only EC technique – in terms of identification accuracy and processing time;

• • Lloyd et al. [Reference Lloyd, Pittiglio, Chandler and Valdastri149] used IPA to optimize the magnetic field of a soft-continuum-magnetic robot such that it conforms to a desired shape by minimizing contact with the environment;

• • Kim et al. [Reference Kim, Ha, Park and Dupont59] used IPA to optimize the curvature sensor placement of a continuum robot by minimizing the shape and tip error between its reconstruction and mechanics-based model;

• • Ros et al. [Reference Ros-Freixedes, Gao, Liu, Shen and Yang38] used multiple runs of IPA (MATLAB’s GlobalSearch) to optimize the joint angle limits of a continuum robot by minimizing the distance between its tip and the possible associated motion plans (i.e., targets);

• • Usevitch et al. [Reference Usevitch, Hammond, Schwager, Okamura, Hawkes and Follmer150] used IPA to optimize the shape of an isoperimetric soft robot by maximizing the volume it encloses; and

• • Ghoreishi et al. [Reference Ghoreishi, Sochol, Gandhi, Krieger and Fuge166] used IPA (i) to implement a hybrid method based on IPA and BO (see Section 3.2.7) for aligning a soft catheter robot to a human blood vessel and (ii) for comparison, showing that their method is more performant.

3.2.2. Levenberg–Marquardt algorithm (LMA)

The LMA [Reference Levenberg134, Reference Marquardt135], or the damped least-squares method, is an iterative gradient-based procedure specifically used to solve nonlinear least-squares problems (i.e., curve fitting) [Reference Levenberg134, Reference Marquardt135]. Being based on gradient descent, LMA cannot be applied to non-differentiable functions.

In our literature review, we found the following research studies that used LMA to optimize the design of soft robots:

• • Tan et al. [Reference Tan, Gu and Ren114] used LMA in a study for the calibration of a soft robot hand, specifically for the kinematics of the finger and the relationship between fingers. They compared LMA with other two techniques: IPA (see Section 3.2.1) and DE (see Section 3.1.3), claiming that both LMA and IPA outperformed DE – the only EC technique – in terms of identification accuracy and processing time.

• • Lai et al. [Reference Lai, Lu, Zhao and Chu152] used LMA to optimize the trajectory of a continuum robot by minimizing the distance between its tip and the target (inverse-kinematics solver).

3.2.3. Steepest descent method

The steepest descent method (SDM) [Reference Curry145], also known as gradient descent, is a gradient-based method that iteratively moves the current solution in the direction of the local downhill gradient.

In our literature review, we found the following research studies that used SDM to optimize the design of soft robots:

• • Chen et al. [Reference Chen, Wang, Li, Chen and Zhu167, Reference Chen, Chen, Cao, Wang, Miao, Gu and Zhu168] used SDM to optimize the skeleton topology of soft-pneumatic network grippers by maximizing the bending angle of the actuator while sustaining prescribed interacting forces.

3.2.4. Nelder–Mead simplex method (NMSM)

The simplex algorithm is a method for linear programming (i.e., a linear objective function subject to linear constraints) [Reference Dantzig174]. It iteratively moves from one feasible solution to another along the edges of a polytope (i.e., a convex region defined by the constraints), selects a pivot element, and performs operations to improve the objective function value until an optimal solution is found. It is a gradient-free method.

While the classic simplex algorithm was designed for linear programming, the NMSM [Reference Lagarias, Reeds, Wright and Wright175] variant can effectively optimize nonlinear functions, as well as non-differentiable problems or problems with discontinuities in the search space. Its main advantages are its independence from the gradient information and its significant improvement of solutions in the first few generations. NMSM is also considered to be a heuristic search because it relies on exploration and adjustment strategies rather than explicit mathematical models to optimize functions. MATLAB Optimization Toolbox offers an implementation of NMSM with the command fminsearch.

In our literature review, we found the following research studies that used NMSM to optimize the design of soft robots:

• • Abbaszadeh et al. [Reference Abbaszadeh, Leidhold and Hoerner31] used the NMSM to find a set of strain gauge parameters of a soft-aquatic robot that would result in the most accurate kinematic calculation (i.e., minimum error between the kinematic computation and the optical sensor measurements);

• • Burgner et al. [Reference Burgner, Gilbert and Webster153] used the NMSM to optimize the volume design of a concentric-tube robot by maximizing the coverage of the required workspace for surgical operations;

• • Bergeles et al. [Reference Bergeles, Gosline, Vasilyev, Codd, del Nido and Dupont154] used the NMSM to optimize the design of a concentric-tube robot by minimizing its length and curvature that can still reach all required paths in a stable manner (i.e., its topology), preferring this algorithm to the PS (see Section 3.2.13) they used in their previous study [Reference Bedell, Lock, Gosline and Dupont161]; and

• • Rucker et al. [Reference Rucker, Jones and Webster155–Reference Rucker and Webster III157] used NMSM to optimize the design of a continuum robot by minimizing the Euclidean distances between the model prediction and experimental data from the perspective of its kinematics, statics, and dynamics.

3.2.5. Constrained optimization by linear approximations (COBYLA)

The COBYLA [Reference Powell141] is an algorithm for nonlinear constrained optimization. It is gradient-free, and it works by iteratively adjusting the values of decision variables using linear approximations of the objective and constraint functions. COBYLA is particularly efficient with noisy and discontinuous objective functions.

In our literature review, we found the following research study that used COBYLA to optimize the design of soft robots:

• • Schiller et al. [Reference Schiller, Seibel and Schlattmann35] used COBYLA to estimate the forward kinematics of a gecko-inspired soft robot by minimizing the energy while simulating its next pose during locomotion.

3.2.6. Cyclic coordinate descent (CCD)

CCD [Reference Martin, Barrientos and Cerro137] is a deterministic algorithm that iteratively updates one decision variable (a coordinate in the search space) at a time while holding the others fixed, cycling through the variables until convergence. CCD can be applied to both minimization and maximization problems and is gradient-free and computationally efficient in problems with easily separable variables.

In our literature review, we found the following research study that used CCD to optimize the design of soft robots:

• • Zhang et al. [Reference Zhang, Wang, Meng, Wang and Liang158] used CCD to solve the inverse kinematics of a soft-continuum manipulator by minimizing the distance between the current position of the robot’s end effector and the target position by moving the robotic joints in turn.

3.2.7. Bayesian optimization

Bayesian optimization (BO) [Reference Frazier142] is a probabilistic and gradient-free optimization method. By exploiting Gaussian processes, it can retrieve optimal solutions without explicit knowledge of the objective function; and as such, it is considered a heuristic search. BO is particularly effective in scenarios where the objective function is complex, costly to evaluate, and lacks a known mathematical form.

In our literature review, we found the following research studies that used BO to optimize the design of soft robots:

• • Ghoreishi et al. [Reference Ghoreishi, Sochol, Gandhi, Krieger and Fuge166] designed a soft catheter for medical applications, which requires finding the best structure that aligns with the vessel shape investigated by the catheter. They used (i) a modified Bayesian technique to seek the optimal geometric and material properties of the soft catheter by minimizing the deviance of the actuated catheter from a desired vessel shape and (ii) IPA (see Section 3.2.1) to optimize the actuator moments for the specific vessel shape.

3.2.8. Method of moving asymptotes (MMA)

The MMA [Reference Svanberg146] is an algorithm for nonlinear constrained optimization – specifically designed for problems with a large number of constraints. The heuristic consists of iteratively keeping track of and adjusting the parameters defining lines and surfaces that approximate the problem’s constraints (the asymptotes). This strategy allows MMA to converge to the optimal solution. MMA is considered a gradient-based optimization method as it involves derivatives in the constraint-handling process. It is claimed to be the most used optimization method for topology optimization [Reference Wang, Zhang, Zhu, Zhang, Chen and Wang169].

In our literature review, we found the following research studies that used MMA to optimize the design of soft robots:

• • Wang et al. [Reference Wang, Zhang, Zhu, Zhang, Chen and Wang169] used MMA to optimize the topology of a soft gripper by minimizing a weighted linear combination of output displacements given external forces; and

• • Li et al. [Reference Li, Chen, Song, Liang, Zhu and Chen41] used MMA to optimize the topology of a soft gripper by minimizing the out-of-plane displacement while also maximizing the in-plane bending deformation of the soft finger under internal pressure and external loads.

3.2.9. Newton–Raphson method (NRM)

The NRM [Reference Kelley147] is a numerical optimization algorithm used for problems with real-valued decision variables. It is gradient-based as it updates the current solution by taking steps proportional to the inverse of the objective function’s second derivative to iteratively converge to the optimum.

In our literature review, we found the following research study that used NRM to optimize the design of soft robots:

• • Morgan et al. [Reference Morgan, Osorio and Arrieta170] used NRM to optimize the design of a soft gripper by minimizing its total energy such that the spring lengths match the required features of the device and facilitate open-loop control.

3.2.10. Greedy algorithms (GRD)

GRD algorithms [Reference Curtis138] are a class of optimization methods that make iterative choices only based on a defined heuristic. The term greedy refers to their approach of immediately selecting the best available option based on the current state, without considering the potential long-term consequences. While not always guaranteed to find the globally optimal solution, GRD algorithms are computationally efficient and straightforward. For example, they are efficient in path-finding problems where agents (or, in our context, robots) explore a free environment with no obstacles, as they directly move toward the target; however, when obstacles are in the environment, they might not consider them in the heuristic and get stuck while their target is right behind the obstacle.

In our literature review, we found the following research study that used GRD algorithms to optimize the design of soft robots:

• • Koehler et al. [Reference Koehler, Usevitch and Okamura10] implemented a GRD algorithm while optimizing the design of a soft 3D haptic shape display, specifically for actuator placement. The algorithm adds actuators iteratively to correct the worst error, which is calculated through control simulation.

3.2.11. Hill climbing (HC)

HC is a local search method that follows the objective function’s direction toward better solutions. In brief, it is the gradient-free equivalent of gradient descend methods (see Section 3.2.3). It explores the search space starting from a random point and then deterministically updates its value by following the most fitting direction of the objective function. This heuristic makes HC very efficient in converging to an optimal solution; however, if the objective function features a non-smooth landscape (i.e., it has many local optima), the algorithm gets stuck in a local optima without the guarantee of finding the global one. Multiple runs of HC might be needed to converge to the global optima.

In our literature review, we found the following research study that used HC to optimize the design of soft robots:

• • Fathurrohim et al. [Reference Fathurrohim, Zuhal, Palar and Dwianto100] combined HC and a GA (see Section 3.1.1) to tune the hyperparameters of GPR (see Section 3.1.7). This hybrid strategy applied to neuroEvolution was used for optimizing the stiffness of a soft-robot fish’s fin by maximizing the time-averaged thrust force it produces.

3.2.12. Simulated annealing (SA)

SA [Reference Kirkpatrick, Gelatt and Vecchi143] is a stochastic local search method for optimization. Its heuristic is inspired by the annealing process in metallurgy, in which the physical and chemical properties of a material are altered by reducing the temperature to increase its ductility (e.g., the process used to manufacture sword blades). Like HC (see Section 3.2.11), SA starts with an initial solution and explores the search space by accepting moves with a probability based on the Boltzmann distribution – that is, it can also accept less fitting solutions. If the next point has a better value, then SA accepts it and reiterates the procedure; if not, SA accepts it with a probability depending on how much worse the point is with respect to the previously generated one and how long the algorithm has been running to simulate temperature (i.e., hot at the beginning and cold toward the end). This allows SA to escape local optima, which is the main downside of HC. However, the cooling simulation should be very slow to enforce regularities of the objective’s layout, resulting in long runs.

In our literature review, we found the following research study that used SA to optimize the design of soft robots:

• • Ghoreishi et al. [Reference Ghoreishi, Sochol, Gandhi, Krieger and Fuge166] used SA for comparison to evaluate their method based on BO (see Section 3.2.7) and IPA (see Section 3.2.1), showing that their method is more suitable for aligning a soft catheter robot to a human blood vessel.

3.2.13. Pattern search (PS)

PS [Reference Audet and Dennis140] is a gradient-free algorithm that explores the search space by modifying the values of decision variables with different patterns. It is considered a heuristic search due to its systematic exploration strategies for exploring the search space, making it well-suited for complex optimization problems having, for example, discontinuous objective functions. MATLAB Optimization Toolbox offers an implementation of PS with the command patternsearch.

In our literature review, we found the following research studies that used PS to optimize the design of soft robots:

• • Bedell et al. [Reference Bedell, Lock, Gosline and Dupont161] used PS to optimize the design of a concentric-tube robot by minimizing its curvature and length for navigation of cluttered environments (specifically human arteries and heart). However, in a follow-up study on the same robot and with the same optimization criteria [Reference Bergeles, Gosline, Vasilyev, Codd, del Nido and Dupont154], the authors defined PS as “computationally inefficient due to its sampling approach” and decided to repeat their optimization using NMSM (see Section 3.2.4).

• • Anor et al. [Reference Anor, Madsen and Dupont162] used PS to optimize the design of a continuum robot by minimizing its complexity (i.e., solved the inverse kinematics retrieving the minimal number of sections composing the robot) in neurosurgical procedures (i.e., specifically designed to work inside the human brain). They minimized (i) the percentage of the robot residing outside the workspace and (ii) the total length of the robot to avoid unnecessary looping or coiling.

3.2.14. Forward and backward reaching inverse kinematics (FABRIK)

FABRIK [Reference Aristidou and Lasenby139] is a gradient-free heuristic algorithm commonly used in computer graphics and robotics to solve inverse kinematics problems. It iteratively adjusts the values of joint angles in a kinematic chain until the end effector reaches the desired position; specifically, it alternates a forward pass in which joints are updated from the base to the end effector and a backward pass in which joints are updated from the end effector to the base.

In our literature review, we found the following research studies that used (or rather, were inspired by) FABRIK to optimize the design of soft robots:

• • Wu et al. [Reference Wu, Yu, Pan and Pei159, Reference Wu, Yu, Pan, Li and Pei160] introduced a heuristic algorithm specifically for solving the inverse kinematics of continuum robots, used for path planning and obstacle avoidance. They named it the Continuum Robot Reaching Inverse Kinematics (CRRIK), and it is based on FABRIK while inspired by the physical process of pulling a rope with a fixed end. This algorithm features a high convergence rate and low computational cost, which are suitable for real-time applications, and it is the most efficient when compared with other inverse-kinematics solvers (i.e., FABRIK [Reference Aristidou and Lasenby139], CCD [Reference Martin, Barrientos and Cerro137], Follow The Leader (FTL) [Reference Brown, Latombe and Montgomery176], or calculating the Jacobian matrix).

3.2.15. Parametric quadratic programming algorithm (PQP)

PQP is a method to solve quadratic programming problems [Reference Best177] that trace optimal solutions on a homotopy path between two quadratic problem instances – that is, when one continuous function can be continuously deformed into another, such that it is possible to move from one topological space to another [Reference Gray178]. This algorithm forms the basis for the algorithms included in the open-source $\text{C}\texttt{++}$ software package qpOASES [Reference Ferreau, Kirches, Potschka, Bock and Diehl144], which outperforms other popular academic and commercial quadratic-programming solvers on many small-to medium-scale convex test examples. PQP and qpOAESIS are not gradient-based methods.

qpOASES is mostly used to optimize the design of soft robots by solving the inverse kinematics while minimizing the displacement of its actuators from the desired position. This was applied to:

• • a cable-driven soft gripper by Adagolodjo et al. [Reference Adagolodjo, Renda and Duriez163];

• • a cable-driven soft robot by Coevoet et al. [Reference Coevoet, Escande and Duriez164]; and

• • a soft-rigid hybrid arm by Coevoet et al. [Reference Coevoet, Adagolodjo, Lin, Duriez and Ficuciello165] .

Specifically, they defined their model as a constrained optimization problem and retrieved the optimal Lagrangian multipliers [Reference Rockafellar179]. We observed that this method is usually associated with robots modeled with finite element method.

3.2.16. Broyden–Fletcher–Goldfarb–Shanno algorithm (BFGSA)

The BFGSA [Reference Nocedal and Wright148] is a gradient-based method for unconstrained optimization. It converges to the optimal solution by iteratively estimating the objective function’s inverse Hessian matrix (a square matrix composed of second-order partial derivatives of the objective function).

In our literature review, we found the following research study that used BFGSA to optimize the design of soft robots:

• • Maloisel et al. [Reference Maloisel, Knoop, Schumacher, Thomaszewski, Bächer and Coros171] used BFGSA to optimize the load-displacement profile and design parameters of a soft-flexible link mechanism by minimizing its force equilibrium equation.