The last decades have witnessed a rapid advancement of robotics applications from well-structured and defined industrial environments into an unstructured and dynamic external environment (Bruzzone and Quaglia, 2012; Rubio et al., 2019). Robots in the real world can move on solid ground as well as in the air and water, and consequently, researchers and engineers have developed terrestrial, aerial, and underwater robots (Aguilar et al., 2016). However, robots still cannot access all types of natural environments (Bruzzone and Quaglia, 2012; Aguilar et al., 2016; Rubio et al., 2019).

Soft yielding substrates are materials that significantly deform under the application of stress and do not recover their original shape when stress is released. These materials can present a yield stress under which they do not undergo plastic deformation, but once pressure exceeds the yield stress, the material flows and undergoes irrecoverable, plastic deformation (Balmforth et al., 2014; Coussot, 2014). Such materials are common in nature: soils are a mixture of different particles such as gravels, sands, silts, and clays (depending on the particle size) mixed with air, water, and organic matter (Barnes, 2016). Mud is such a medium with high water content. Sand, snow, and leaf litter are other examples of such materials with varying properties (Wong, 2009; Balmforth et al., 2014; Barnes, 2016).

The problem of locomotion in deformable grounds is important to solve because it would facilitate new robotic applications such as search and rescue in wet forests, muddy fields, avalanches, and mudslides (Schneider and Wildermuth, 2016); for the agricultural vehicles operating on wet soils (e.g., rice fields) (Duckett et al., 2018; Oliveira et al., 2021); for exploration or excavation of materials (e.g., wood and ore) with a minimal environmental impact (Billingsley et al., 2008; Lopes et al., 2020); for environmental monitoring in high-biodiversity areas (e.g., river estuaries, bogs, and shores) (Dunbabin and Marques, 2012); or for extra-terrestrial exploration (Li and Lewis, 2022).

The field of terramechanics (Bekker, 1960; Wong, 2009) covers interactions between vehicles and natural grounds from a traction perspective. It covers the theory of vehicles moving with wheels or tracks on mud, sand, or soil. However, as we demonstrate in this article, there are very different ways of moving on soft deformable grounds, and terramechanics covers only one of these, which is the method used by all wheeled and tracked vehicles.

Well-known manned vehicles and some robots use wheels and tracks to move in these environments up to a limit (Bruzzone and Quaglia, 2012). Usually, they are large and heavy enough to deform the medium and gain traction from the solid bottom under the loose substrate if the medium is non-homogeneous and from the unyielded buried substrate, otherwise. Yet, this can fail if the medium is too deep and weak because actuators cannot generate sufficient tangential forces to move a potentially buried body. Some robot prototypes are addressing the challenge of traversing yielding terrains. Some quadruped (Raibert et al., 2008; Bagheri et al., 2017) or hexapod robots with rigid (Li et al. (2013); Li et al. (2009)) or adaptable legs (Liang et al., 2012) have been shown to be capable of negotiating those terrains. The SeaDog robot uses spoke wheels, which combine some advantages of wheels and legs (Klein et al., 2012), and the ePaddle robot uses wheels with expandable paddles (Shen et al., 2018). Undulatory robots mimicking worms (Sfakiotakis et al., 2016), snakes (Marvi et al., 2014), or lizards (Maladen R. D. et al., 2011) are other examples. Unstructured environments have also been negotiated by crawling robots such as a sea-turtle robot (Mazouchova et al., 2013) and a mudskipper robot (McInroe et al., 2016). A razor clam robot was designed, capable of digging through mud (Winter et al. (2014)). Some researchers designed screw-based robots, having either two screws (Nagaoka et al., 2010b) or four (Lugo et al., 2017). Recently, some robots were designed to challenge the granular lunar terrain (Shrivastava et al., 2020) by combining wheels and walking gaits. On a different scale, a sperm-inspired robot has been built, moving at low Reynolds numbers (Khalil et al., 2016). Aguilar et al. (2016) introduced the field of robophysics which consists of studying the motion of moving systems by complementing the study of complex robots with simplified robotics experiments and simple theoretical models. It provides a review of research that has already helped with the understanding of unstructured environments.

Bioinspiration and biomimetics seems to be the most commonly used paradigm for developing robots for yielding environments: the majority of the aforelisted examples explicitly claim to be inspired by animal locomotion. Typically, they mimic a specific animal or an aspect of the locomotion of a specific animal. Bio-inspired robotics can be used in two ways: one can take inspiration from a known working solution taken from nature to make a robot particularly well-suited for an environment, thereby benefiting robotics. It can also be used the other way around: one can make a robot that mimics a living being to understand some of its working principles, thereby benefiting biology (Gravish and Lauder, 2018). The concept of biomimetics and bioinspiration is a specific case of a problem-solving technique called design-by-analogy (Verhaegen et al., 2011) and often has its focus on biomechanics (Goel et al., 2017). The analogy between biology and engineering can be derived at different levels, but the decisive phase is the mapping from the problem domain to the solution domain using the most appropriate level of abstraction (Vincent et al., 2006; Verhaegen et al., 2011). It is critical to find an appropriate abstraction level for the task when using biomimetics (Nagel et al., 2010), and a higher level of abstraction allows for the creation of links between different principles that are independent of the form or behavior of biological entities (Mak and Shu, 2004). When abstract principles have been derived, they can be transposed to technology and the abstraction level can be lowered to reach technological solutions for which direct mimicry of biology–technology may not be feasible (Baumeister et al., 2013; Fayemi et al., 2017).

A majority of the robotics literature we analyzed chose to mimic the anatomy of an animal, its locomotion, or a feature of its body or behavior. Through mathematical modeling, some researchers have begun to unify these principles of locomotion in dry granular media. For example, Astley et al. (2020) reviewed the principles and mathematical modeling of limbless locomotion in dry sand, Zhang and Goldman (2014) demonstrated the applicability of resistive force theory (RFT) in dry granular media, and Hosoi and Goldman (2015) proposed a mathematical modeling technique for digging and burrowing in granular media and described four regimes based on the size of the animal and the inertial number. Other works also address different media. For example, Dorgan (2015) established the mechanisms of subterranean locomotion in dry and cohesive media, and Aguilar et al. (2016) proposed using mathematical modeling, simulations, and experimental validation as a systematic approach to locomotion in a variety of environments such as air, water, hard ground, or cluttered environments and goes beyond the field of yielding media. There, locomotion in yielding materials is treated from an anatomical point of view and describes the modes of locomotion as a legged, flipper-based, sand-swimming, or two-anchor mechanism.

The very active work on modeling presented previously is a necessary effort to understand the mechanisms of locomotion in yielding media. However, while reviewing the research work on biology, we found that, in the vast majority of cases, mathematical modeling of the animals’ locomotion was not available. To enable their study, we propose turning to a higher abstraction level where locomotion can be studied from a qualitative point of view, such as the higher level prescribed by the biomimetics methodology.

In this paper, we review biology literature addressing locomotion in soft, deformable environments and propose some general principles for designing robots for those terrains. We use abstraction at the level of general mechanical principles of the deformable medium to derive engineering goals. We also categorize the existing robotics literature and demonstrate that some of those principles have been explicitly or implicitly already used in robots or other vehicles, and since they are sufficiently general, the resultant design does not necessarily need to be explicitly bio-inspired. This overview aims to offer a systematic approach, general guidelines, and design targets for developing better vehicles for soft yielding environments. The higher abstraction level used in this overview enables drawing parallels between locomotion strategies that may, at first, seem distant from each other. As discussed previously, the use of a high abstraction level such as the one we chose is advocated by the biomimetics methodology. With this paper, we aim at providing future researchers in this field with a tool to pass from the problem domain to the solution domain and find an adapted solution for their application related to locomotion in yielding environments. In addition to this engineering tool, we believe this classification can aid in the discovery of links between locomotion strategies and provide hints for future research into the unification of locomotion theories in soft yielding materials. Last, the present paper will be a useful catalog of possible sources of inspiration for locomotion in soft media. The principles and strategies of locomotion presented in this paper could be further modeled and experimented with using and extending the theories presented previously, for example, through the robophysics framework.

Yielding materials are a subset of non-Newtonian fluids, which are fluids that exhibit shear stress that is not proportional to the shear rate. Over the last century (Alderman (1997); Denn (2004) for overviews), researchers have identified a wide range of complex behaviors exhibited by non-Newtonian fluids, including shear-thinning, shear-thickening, Bingham plastics, and viscoplastic fluids (Bingham (1917; 1922); Herschel and Bulkley (1926); Oldroyd (1947); Metzner and Reed (1955); Metzner (1956); Schowalter (1960)). Rheology, which is the study of the flow and deformation of matter, is a fundamental discipline that bridges the study of non-Newtonian fluid mechanics and the theory of plasticity (Barnes et al., 1989; Larson, 1999). It provides a framework for understanding and quantifying the complex behaviors of non-Newtonian fluids, as well as other complex materials beyond fluids, such as polymers, gels, pastes, and muds. This understanding of complex materials is essential to comprehend the behavior of soil, which is a complex substrate made up of four components: minerals, air, water, and organic matter. These components can be present in varying proportions depending on precipitation, proximity to water bodies, or compression of the material. Depending on the particle size of minerals, the soil particles can be classified as clay (smallest), silt, sand, or gravel (largest). When the water content is very high, the soil behaves as a liquid. When the water content decreases, the soil behaves as a plastic or viscoplastic material and is solid when the water content is low (Barnes, 2016). The plastic behavior means that above a certain stress, called yield stress, the material deforms and does not return to its original shape. The smaller the particles, the lower the permeability to water, and the more prominent the plastic behavior is.

The stress applied on the material can be computed with the von Mises criterion (1): σ = σ 1 − σ 2 2 + σ 2 − σ 3 2 + σ 3 − σ 1 2 2 , (1) where σ _i denotes the stresses in each principal direction and is calculated as in Eq. 2, and σ denotes the von Mises stress. Above a certain value called “yield stress,” the material will undergo plastic deformation and flow. For non-cohesive materials like sand or dust, the yield stress is negligible, and the material starts to flow as soon as it is pressed upon. σ i = F i S i . (2)

In Eq. 2, F _i is the force in direction i, and S _i is the contact area in the same direction. Forces can be either due to the weight of the body or due to acceleration.

Viscoplastic behavior exhibits both solid and fluid properties, and material deformation is also affected by the rate of stress (Kutter and Sathialingam, 1992; Balmforth et al., 2014; Coussot, 2014). A large selection of models exists for muddy/clayey/sandy soils, but a unifying model is still to be found (Liingaard et al., 2004; Karim and Gnanendran, 2014). Soft yielding grounds are manifold and can present different properties, depending on whether they are cohesive, such as mud and wet sand, or not cohesive, such as dust or dry sand. More specifically, contrary to non-cohesive yielding grounds, cohesive grounds have a yield stress and are viscous (Augustesen et al., 2004; Omidvar et al., 2012; Mishrai et al., 2016; Yang et al., 2017).

To move, a body must exert forces on the environment. The environment, in turn, generates a counter-reaction force that propels the body. On a solid flat ground, if the body stands still, the force on the ground, called the ground reaction force, has only a normal component to compensate for the weight of the body. When the body moves, it must exert a force parallel to the ground, which is the horizontal component of the ground reaction force and is limited by the friction coefficient of the ground/body couple. When the ground is soft, it can deform under the weight of the body, and the maximum force is also limited by its shear resistance. The deformation of the material causes energy losses that make locomotion more difficult as more work is required (Lejeune et al., 1998). The work W exerted on a deforming medium is defined as in Eq. 3, where z is the depth of intrusion and F the force as a function of z. W = ∫ F d z . (3)

The body will move forward when it generates sufficient forces F in the direction of movement over some time t (mechanical impulse I) to move its mass m at a certain speed v (to gain momentum): I = ∫ F d t = m Δ v . (4)

When the soil is yielding, the applied force is limited by the yield force in the direction of movement, reducing the impulse and slowing the body down. To reduce sinkage into the media, robots or animals have to apply reduced pressure. When sinkage already occurred, the animals or robots have to deal with moving in a deformable material, where, depending on the material, stress, rate of stress, and depth of insertion can all play a role (Terzaghi et al., 1996).

Given that soft yielding materials exhibit different behaviors depending on how one interacts with them, there are two different ways of moving in/on them. When moving on a yielding substrate, frictional hydrostatic forces dominate at low speeds, while the inertial hydrodynamic response of the material dominates at high speeds (Qian et al., 2012). There are, thus, two ways to move on or through a yielding material: using the material’s static properties or using its dynamics properties (Figure 1).

There are primarily two ways to move using the static properties of the medium, as shown in the top part in Figure 1. The first way is to not exceed the yield stress of cohesive materials or deform them plastically so that the resulting surface can be used as a static support against which to generate thrust. When we do not consider dynamics, the vertical equilibrium is maintained between hydrostatic-like pressure on the immersed volume of the body and weight. We refer to hydrostatic-like pressure as a pressure that increases proportionally with depth (Aguilar and Goldman, 2016). It has been shown that different flowable materials, when stepped upon, generate a reaction force that is almost proportional to the depth of intrusion (Sharpe et al., 2013; Godon et al., 2022; Ma et al., 2022). In the horizontal direction, thrust can be generated by relying on the cohesion and adhesion of the medium particles with the body. Additionally, if plastic deformation is carried out, the body can push horizontally against the unveiled surface of the material.

In the second case, the moving body uses the dynamic fluid properties of the medium (see the bottom part of Figure 1). To move dynamically on/in a fluid, a body has to transfer momentum to the fluid so that the fluid reacts by providing thrust, as per the principle of the conservation of momentum. There are three main momentum transfer mechanisms in fluids: drag (pressure and friction drags), lift, and acceleration reaction forces. However, in friction-dominated media, lift cannot be used for propulsion (Webb, 1988), and pressure drag is another manifestation of inertial forces (Vogel, 2020). This leaves two mechanisms for movement through a frictional fluid. The first, friction drag, can be used in static locomotion and is also a necessary component of pressure-drag-induced inertial forces. The second, acceleration-related force, takes advantage of propelling fluid particles in the opposite direction of the desired thrust. In nature, this is usually performed in the fluid by oscillations or undulations of the body and appendages in fish, ducks, or seals (Sfakiotakis et al., 1999; Vogel, 2020), but can also be performed at the surface by slapping the fluid. In this rapid motion regime, the forces exerted by the fluid depend quadratically on velocity. We refer to this as the hydrodynamic-like principle (Aguilar and Goldman, 2016) because of its similarities to the hydrodynamic principles of propulsion exerted in fluids (Webb, 1988; Sfakiotakis et al., 1999; Vogel, 2020).

As opposed to moving on a deformable soft media, dynamic motion through such medium is hindered by high friction drag and form drag. One solution to reduce drag is to use fluidization. Fluidization is the process by which material particles are given a sufficient velocity so that the granular material (cohesive or not) behaves as a fluid. This can be performed by injecting a fluid through the materials, which will generate a form drag on the particles and reduce the interparticle stress. Fluidization can also be enhanced by vibrations in a granular material (Zik et al., 1992; Xu and Zhu, 2006). Fluidization is mostly used to move through the substrate to facilitate progression and is hence typically observed in animals digging or burrowing in granular media (Hosoi and Goldman, 2015).

Thus, locomotion on non-Newtonian yielding ground is different from that on solid ground or in water. To move on/through such materials, animals have developed a wide range of locomotion strategies. These strategies are explored and categorized in this paper, and existing robots are categorized accordingly.

This paper aims at answering the following questions:

- How to classify the modes and mechanisms of locomotion in natural, yielding environments?

- What physical principles are present in nature for locomotion in yielding environments?

- Do those modes and mechanisms share any common physical principles that are abstract enough to be applied to robots including non-bio-inspired robots?

To answer these questions, we conducted a review of biology research, focusing at first on animal locomotion in soft yielding environments. We identified relevant research papers and classified them based on material behavior and locomotion mechanisms. We have also mapped existing robots into the presented principles, demonstrating that successfully developed solutions are designed implicitly or explicitly against those principles and suggesting that defining those principles as design targets will help develop better vehicles faster and easier.

Our integrative literature review on the biology of locomotion in deformable environments used established methodologies outlined in Torraco (2005); Snyder (2019). We utilized relevant keywords associated with the themes of locomotion, animals, and soft environments (Table 1) to search various online databases such as Google Scholar, IEEExplore, ACM Digital Library, Science Direct, Web of Science, Wiley Online Library, Scopus, CiteSeerX, SpringerLink, PNAS, and PLOS One. Our search terms involved combining at least one word from the first column with at least one word from the second column in Table 1 to create different keyword combinations (e.g., “animal locomotion in multiphase environments,” “walking fish on mud,” and “legged locomotion on low resistance ground”). Additional papers were identified from the references in the documents. To ensure inclusiveness, all papers mentioning any of the selected keywords were selected at first. Then, analysis of the main topic of the documents led to keeping only those addressing locomotion from a mechanical perspective. For instance, documents discussing the evolution of the genome, neural control, muscle control, fish swimming near the surface, or an evolutionary analysis of anatomy were excluded. The documents were then divided into categories based on the main topic, and some categories were disregarded as being out of scope for this literature review (e.g., addressing the effect of viscosity on swimming or the burrowing patterns of crustaceans). The most representative documents of each category constitute the corpus of this article and were organized based on the mechanical principles and locomotion strategies employed by the animals they describe. Papers relating to robotic analogs were included at a later stage.

We, then, analyzed the locomotion used by each species and robot and abstracted it to understand how and where the forces are generated. We discovered that even when forces were generated using different body parts, motions, speeds, and material properties, they could be classified into two categories based on whether they used the static or dynamic properties of the yielding media. Then for each mechanical principle, the interaction type could be generated using different strategies, which mainly differed depending on the nature of the contact between the body and the medium. Furthermore, under this second level, we discuss the animals’ specific means of locomotion, describing which body parts are used, how they are used, and the special features facilitating the means of locomotion.

In the next sections, the locomotion strategies classified according to the mechanical principle will be more closely described, along with a description of the specific animal locomotion patterns and references to the existing robotic analogs.

In the following sections, the animals’ and robots’ means of locomotion on soft yielding grounds will be categorized into the two following interaction types: those using the static properties of the material, mainly using friction-based hydrostatic pressure, and those using the dynamic properties, mainly inertia.

Statics-based movements use the hydrostatic-like pressure in the medium to counter gravity, friction forces, and material cohesion to generate forward impulse. We observed that the highest level of distinction that could be made was based on the spatial distribution of the contact between the body and the media. Three different strategies were observed:

• Discrete contacts with the medium

The two first strategies imply compensating the body’s vertical and forward momentum on the ground by achieving static equilibrium between the deformable material and the body. There are two possibilities: the stresses exerted by the body on the medium are less than or equal to the medium’s yield stress. In most cases, unless the animal is very light-weighted, or the ground is almost solid, the material first yields, and then solidification occurs when pressure drops below the yield stress: at first contact, only a minor portion of the body, for example, a foot, touches the ground, and the pressure is very high. The surface yields, the foot sinks into the soil until a sufficient surface area touches the substrate to distribute the efforts, and the relative pressure is reduced to the yield stress. Resistive force increases linearly with depth: the deeper the appendage sinks into the substrate, the higher the pressure, and the more the substrate resists intrusion. Only when solidification has occurred, the animal can use the substrate as a static support. Yet, at this stage, the material has deformed, required work, and made locomotion harder. There are two different possibilities to reduce sinkage: reduce the pressure exerted on each weight-bearing body part and reduce acceleration-related forces.

Animals use different strategies to achieve such results: using a large number of legs, using large feet surface areas, lying on a large portion of the trunk, using the tail to increase the contact area, or reducing forces by moving slower. In the third strategy, through the medium, the body is inside the substrate, and hydrostatic-like pressure does not need to compensate for the weight of the body outside the material. There, the challenge is rather to generate forces using non-reciprocal movements because the friction-dominated environment dissipates energy, and therefore inertia (Purcell, 1977; Vogel, 2020).

To move using the dynamic properties of the medium, a body has to rely on the medium’s inertia to exert a sufficient force. To facilitate locomotion on/through a yielding material, a body can make it behave as a fluid, thereby reducing its resistance. Analogous to what was observed in the statics-based interactions, two different strategies were observed:

• Discrete contacts with the medium

We did not find any examples of dynamics-based locomotion using continuous contact with the medium in the robotics or biology literature. However, some man-made vehicles use this strategy. Examples are a full-throttle dirt bike going over a mudflat or a propeller-propelled boat powerful enough to generate thrust in mud. Although the reason for the absence of this strategy in nature or robotics could not be identified in the literature, we believe it does not provide any benefits that would justify its usage. As we will see in this section, the use of dynamics-based motion requires making use of the medium’s inertia with discrete, powerful strokes or fluidizing it. A dynamic locomotion based on continuous contact may not benefit from form drag reduction with fluidization since the major part of the body lies outside of the yielding material. This type of locomotion may not benefit from having a continuous, frenetic movement at the interface between air and yielding media either, as this would likely require substantial power to continuously expel material backward/downward. While animals sometimes use power-intensive locomotion strategies for mating, escaping, or preying, they generally tend to use energy-efficient locomotion strategies (Schmidt-Nielsen, 1997; Alexander, 2003). It is, thus, not surprising that no animal was discovered using this inefficient strategy nor that no bio-inspired robot was discovered using it, or any robot, particularly given the generally inefficient locomotion of robots (Kashiri et al., 2018).

The locomotion mechanisms described in previous sections often share common physical principles that facilitate the animal to negotiate yielding terrains. Those principles can be used by animals regardless of their anatomy or locomotion pattern and are, therefore, common to many species. A summary of this classification can be found in Table 2. Next, we classified the robots accordingly. This can be found in Table 3.