Cockroaches traverse crevices, crawl rapidly in confined spaces, and inspire a soft, legged robot

Kaushik Jayaram and Robert J. Full
PNAS February 23, 2016 113 (8) E950-E957; first published February 8, 2016 https://doi.org/10.1073/pnas.1514591113

  1. Edited by Z. Jane Wang, Cornell University, Ithaca, NY, and accepted by the Editorial Board December 29, 2015 (received for review July 23, 2015)

Significance

Cockroaches intrude everywhere by exploiting their soft-bodied, shape-changing ability. We discovered that cockroaches traversed horizontal crevices smaller than a quarter of their height in less than a second by compressing their bodies’ compliant exoskeletons in half. Once inside vertically confined spaces, cockroaches still locomoted rapidly at 20 body lengths per second using an unexplored mode of locomotion—body-friction legged crawling. Using materials tests, we found that the compressive forces cockroaches experience when traversing the smallest crevices were 300 times body weight. Cockroaches withstood forces nearly 900 times body weight without injury, explaining their robustness to compression. Cockroach exoskeletons provided inspiration for a soft, legged search-and-rescue robot that may penetrate rubble generated by tornados, earthquakes, or explosions.

Abstract

Jointed exoskeletons permit rapid appendage-driven locomotion but retain the soft-bodied, shape-changing ability to explore confined environments. We challenged cockroaches with horizontal crevices smaller than a quarter of their standing body height. Cockroaches rapidly traversed crevices in 300–800 ms by compressing their body 40–60%. High-speed videography revealed crevice negotiation to be a complex, discontinuous maneuver. After traversing horizontal crevices to enter a vertically confined space, cockroaches crawled at velocities approaching 60 cm⋅s−1, despite body compression and postural changes. Running velocity, stride length, and stride period only decreased at the smallest crevice height (4 mm), whereas slipping and the probability of zigzag paths increased. To explain confined-space running performance limits, we altered ceiling and ground friction. Increased ceiling friction decreased velocity by decreasing stride length and increasing slipping. Increased ground friction resulted in velocity and stride length attaining a maximum at intermediate friction levels. These data support a model of an unexplored mode of locomotion—“body-friction legged crawling” with body drag, friction-dominated leg thrust, but no media flow as in air, water, or sand. To define the limits of body compression in confined spaces, we conducted dynamic compressive cycle tests on living animals. Exoskeletal strength allowed cockroaches to withstand forces 300 times body weight when traversing the smallest crevices and up to nearly 900 times body weight without injury. Cockroach exoskeletons provided biological inspiration for the manufacture of an origami-style, soft, legged robot that can locomote rapidly in both open and confined spaces.

The emergence of terradynamics (1) has advanced the study of terrestrial locomotion by further focusing attention on the quantification of complex and diverse animal–environment interactions (2). Studies of locomotion over rough terrain (3), compliant surfaces (4), mesh-like networks (5), and sand (6⇓–8), and through cluttered, 3D terrain (9) have resulted in the discovery of new behaviors and novel theory characterizing environments (10). The study of climbing has led to undiscovered templates (11) that define physical interactions through frictional van der Waals adhesion (12, 13) and interlocking with claws (14) and spines (5). Burrowing (15, 16), sand swimming (17), and locomotion in tunnels (18) have yielded new findings determining the interaction of bodies, appendages, and substrata.

Locomotion in confined environments offers several challenges for animals (18) that include limitations due to body shape changes (19, 20), restricted limb mobility (21), increased body drag, and reduced thrust development (22). Examining the motion repertoire of soft-bodied animals (23), such as annelids (19), insect larvae (24), and molluscs (25), has offered insight into a range of strategies used to move in confined spaces. Inspiration from soft-bodied animals has fueled the explosive growth in soft robotics (26⇓⇓⇓–30), which is not only creating new perspectives in robot design and control, but also more sharply defining the advantages and disadvantages of soft systems capable of maneuvering in constrained environments. As pointed out by Kim et al. (26), “soft materials also lend themselves to highly flexible and deformable structures, providing additional functional advantages to animals, such as enabling entrance into small apertures for shelter or hunting.” However, disadvantages include weight support against gravity, body/appendage control, and “high deformability and energy-absorbing properties of soft tissues prevent[ing] them from exerting large inertial forces and limit[ing] how fast soft animals can move from place to place” (26).

We explored the capability of cockroaches to not only exploit the advantages of appendages with structures that allow effective interaction with diverse terradynamic surfaces, but that also permit them to function as exceptional soft-bodied locomotors in confined spaces. Cockroaches use rigid, jointed exoskeletons to run rapidly at speeds approaching 1.5 m⋅s−1 (31), climb up walls (11, 32), race along ceilings (33), and swing stealthily under ledges out of sight (34). However, materials science has revealed that the stiffness of exoskeletal tissue can differ by eight orders of magnitude (35⇓⇓–38), permitting the possibility that cockroaches with powerful propulsive appendages might also retain the advantages of soft-bodied animals capable of conforming to their environment (39). As yet, no study has quantified cockroaches’ capacity to traverse crevices and crawl in confined spaces.

We selected the American cockroach, Periplaneta americana, because of its high speed (31), maneuverability (34, 40), robustness (41), and tenacity to enter and leave spaces. To determine the smallest horizontal crevice that a cockroach could traverse, we challenged animals with a series of decreasing crevice heights. We hypothesized a minimum height based on compression tests of body segments.

Once cockroaches entered the confined space and became more sprawled, we hypothesized that running velocity, stride length, and stride period would decrease with crevice height, whereas slipping and the probability of a zigzag path would increase. We used high-speed movie to quantify the kinematics of crawling, postural changes of sprawl angle, and foot or tarsal position relative to free running (31, 42). Motivated by studies of the role of body friction during undulatory swimming in frictional fluids by sandfish lizards (17, 43) and the thrust produced by flipper-driven surface locomotion by sea turtle hatchlings (8), we varied ceiling and ground friction to determine their effect on confined-space crawling performance. Given these results, we created a model of this unexplored mode of locomotion in which drag acts on an animal’s dorsal and ventral surface, but the media does not flow around the animal.

Finally, we hypothesized that compression of the body and legs would demonstrate nonlinear, viscoelastic behavior (41), suggesting crevice crossing might be affected by rate and that the magnitude of peak compression forces would reveal the extent of exoskeletal robustness. As a first step toward quantifying the exoskeletal material properties (41) and shape changes that enable cockroaches to traverse crevices and crawl in confined spaces, we measured the compression of selected head and body segments by adding loads to anesthetized animals and performed a series of dynamic compressive cycle tests on living animals.

Inspired by the data on cockroach segment and body compression, postural change, and kinematics, we designed a legged robot using the smart composite microstructure (SCM) manufacturing (44⇓–46) approach involving laser-cutting, laminating, and the folding of exoskeleton-like plates. We see this robot useful both as a physical model to test future hypotheses of the mechanisms permitting confined-space locomotion, as well as a first step toward the development of a soft search-and-rescue robot that can penetrate the rubble left by tornados, earthquakes, or explosions.

Results and Discussion

Crevice Traversal.

The American cockroach (0.76 ± 0.16 g; 31.10 ± 2.16-mm length; 12.52 ± 1.48-mm height), Periplaneta americana, traversed horizontal crevices as small as 3 mm, the height of two stacked US pennies (Fig. 1 A and B). A vertically adjustable acrylic gate mounted on a precision linear stage permitted control of crevice height (±0.05 mm; Fig. 1B, SI Methods, and Fig. S1A). To set experimental crevice height, we directly measured body compression of freshly anesthetized animals from their abdomen to head while positioned on a flat surface by suspending a load greater than 100 times body weight across each section (Fig. 1C, Fig. S1C, and Table S1). Exoskeletal sections were all highly compressible, ranging from 41% to 57% of normal with the load resulting in no injury.

Fig. 1.
Fig. 1.

Performance of cockroaches traversing horizontal crevices. (A) Comparison of freestanding height of American cockroach, Periplaneta americana, relative to the near-minimum crevice height traversed equal to 3 mm, two stacked pennies. (B) Crevice traversal apparatus with cockroach about to enter (Fig. S1A). (C) Body compression (white vertical bars) resulting from a 100-g load across segment. Percent body compression shown below segment (Fig. S1C and Table S1). (D) Crevice traversal stages extracted from high-speed movie frames with corresponding time stamp for 3-mm height (Movie S1). (E) Crevice traversal time at three crevice heights. Each behavioral stage duration is stacked onto the next from left to right to also show total time. Points and error bars represent mean ± 1 SD. (F) Probability of crevice traversal for three crevice heights (represented by three colors). Cockroaches successfully traversed the greatest heights more frequently and failed to traverse the lowest crevice heights by turning back or getting stuck within the crevice. Number of trials is shown above bar.

Fig. S1.
Fig. S1.

Experimental setup for collecting data and robot schematics. (A) Crevice traversal setup featuring a clear acrylic tube with adjustable opening. (B) Confined-space crawling setup featuring acrylic track with an adjustable ceiling plate and fixed ground plate for determining the effect of gap size. To measure the effect of friction, the ceiling and ground plates were fixed 4 mm apart and surfaces were varied with different grades of sandpaper. (C) Exoskeletal compression process with an anesthetized cockroach and 100-g weight in unloaded (Left) and loaded (Right) configurations.

View this table:
Table S1.

Morphometric measurements of individual cockroaches

Crevice traversal was rapid (288- to 821-ms interquartile range) and, to the naked eye, appeared continuous. High-speed videography revealed crevice traversal to be a complex behavior involving several stages (Fig. 1D and Movie S1). We defined these stages to enable measurement of the time taken to progress through each stage at the three crevice heights. (i) Exploration and crevice detection: Once placed within the apparatus (Fig. 1 B and D), the animals used their antennae to explore their surroundings and searched for escape openings. If one or both antennae passed through the crevice, the animal reoriented itself to ensure both antennae were extending out of the crevice and continued tactile exploration. We suspect this searching behavior enabled the cockroaches to estimate the size of the crevice and to determine whether the other side was safe for escape. (ii) Head traversal with entry and body orientation: On completion of crevice detection, the cockroach typically paused briefly and rammed into the crevice head-first multiple times, often hitting the gate, before pitching downward to the enter the crevice. For larger crevice sizes (6 mm and above), this would be immediately followed by abdominal traversal with little body compression. During this phase, the legs rapidly cycled attempting to grip surfaces and generate sufficient friction. In concert with head entry, the front legs stretched outside the crevice attempting to pull the body through. At the same time, the remainder of body pitched upward, allowing the middle and hind legs to push off the side/top walls of the chamber. (iii) Pronotum traversal: As the leg movements began getting restricted (especially the front legs), the body rapidly rolled from side to side accompanied by leg pushing. The rapid body rotations possibly helped to reorient the middle and hind legs for maximizing thrust production (47) and to potentially passively align the body with respect to the crevice opening. (iv) Thorax traversal: We hypothesized this to be one of most challenging stages with the highest chance of failure by getting stuck due to the body morphology (Table S1). The movement of the legs was severely restricted, and the animal was likely experiencing large normal forces. Animals made forward progress primarily by thrusting their hind legs. (v) Abdomen traversal: After thorax traversal, the body was again reoriented into a “flat” position (body pitch of 0°). The front legs and middle legs were free to operate, whereas hind-leg movement was restricted, making progression difficult. The compressibility of the abdomen appeared to reduce the normal load on the body, enabling the animal to generate thrust sufficient to successfully negotiate the crevice.

The time taken from head entry until abdomen tip exit increased with a decrease in crevice height from 6.1 to 3.2 mm [ANOVA, P < 0.001, F(2,94) = 88.9; Fig. 1E]. Traversal time was similar for 6.1- and 4.4-mm crevices (0% and 27 ± 3% abdomen compression, respectively; post hoc analysis, Tukey’s honest significant difference), but significantly longer at 3.2 mm (47 ± 3% abdomen compression), approaching the limit of performance. Consistent with this conclusion, the probability of successful crevice traversal decreased significantly with a decrease in crevice height from 72 ± 5% at 6.1 mm to just 17 ± 3% at 3.2 mm [Cochran–Mantel–Haenszel, P < 0.001, χ2(2) = 44.8; Fig. 1F]. Turning back during a trial was the dominant failure mode, suggesting that animals seek alternate routes if crevices are too small, presumably using sensory feedback (48). Animals were never trapped in the largest crevices, but occasionally became stuck at the smallest crevice heights (9 ± 2%) during thorax traversal, a potentially fatal event in nature if exposed to predators.

Confined-Space Crawling.

After traversing narrow crevices, cockroaches crawled rapidly in spaces confined vertically by two stacked horizontal plates (SI Methods and Fig. S1B) at velocities approaching 60 cm⋅s−1, despite body compression and large postural changes such as sprawl angle (Movie S2). High-speed kinematic analysis of marked animals at four ceiling heights (Fig. 2 A and B) revealed that velocity [58.05 ± 2.33 cm⋅s−1, ANOVA, P = 0.56, F(2,72) = 4.72] did not vary significantly across the three greatest ceiling heights, and only decreased at the smallest height where animals experienced the greatest body compression and friction [4 mm; 14.52 ± 0.98 cm⋅s−1; ANOVA, P < 0.001, F(3,96) = 14.72; Fig. 2C]. Effective stride length explained the changes in velocity, whereas stride period remained constant across all crevice sizes [61.05 ± 6.06 ms, P = 0.34, F(3,96) = 1.28]. Ceiling height constrained posture by increasing sprawl angle proportionally [P < 0.001, F(3,96) = 24.12; Fig. 2B]. Surprisingly, the distance from the foot to the body midline, as viewed from the top (tarsus midline distance), did not change significantly across ceiling heights [11.89 ± 0.96 mm; P = 0.64, F(3,96) = 3.61; Fig. 2C]. This foot placement is similar to that most effective for maximizing ground thrust as predicted from cockroach musculoskeletal models (49). Cockroaches used an alternating tripod gait at all ceiling heights except the smallest where the gait became irregular with the middle legs often applying force in synchrony to thrust the body forward. Consequently, at the smallest ceiling height, animals were unable to follow straight-lined paths [decreased tortuosity index; P = 0.002, F(3,96) = 5.49; Fig. 2C]. Foot slippage also increased at the smallest ceiling height [decreased stride success ratio; P = 0.005, F(3,96) = 3.31; Fig. 2C]. The ability of cockroaches to maintain performance with minimal changes in kinematic parameters at the three greatest ceiling heights is similar to outcomes in experiments with ferrets running in tunnels with back height reduced by 40% and hip height by 25% (50). However, at the smallest ceiling heights, results for confined-space crawling suggest that cockroaches were attempting to generate sufficient thrust through ground engagement, but because of body contact with the ground and ceiling were operating in a “friction-limited” environment. Under these conditions, thrust on the body is expected to equal the maximum friction force offered by the surface because of foot slippage.

Fig. 2.
Fig. 2.

Confined-space crawling performance of cockroaches. (A) Crevice crawling apparatus with cockroach about to enter (Fig. S1). Ceiling heights used represent freestanding (12 mm), crouched (9 mm), just beginning to compress body (6 mm), and minimum ceiling height within which animals crawled (4 mm). (B) Side (from movie) and front view of cockroach crawling within chamber at two ceiling heights. Front view shows the increase in sprawl angle, but not foot-to-body midline distance (tarsus midline distance) as ceiling height was reduced. (C) Performance metrics, velocity (gray), stride length (red), stride period (magenta), sprawl angle (green), tarsus midline distance (dark blue), stride success ratio (ratio of successful strides with no foot slipping relative to the total number of strides; light blue), and tortuosity index (forward displacement of cockroach relative to the length of the actual path taken; orange) as a function of ceiling height with their respective units are indicated in parentheses after label. Points and error bars show mean ± 1 SD. Red stars represent a significant difference at 4 mm relative to larger ceiling heights.

Body-friction legged crawling operates in a unique regime where drag is exerted on the body, friction dominates thrust, but the media does not flow. To determine whether cockroaches undergo confined-space, body-friction legged crawling, we varied the ground and ceiling friction of the apparatus (Fig. 3). We directly measured the coefficients of friction using anesthetized animals interacting with different surfaces to produce three levels of friction for both the ceiling (fivefold change) and ground (twofold change; SI Methods and Table S2). We hypothesized that increased ceiling friction and dorsal drag would decrease velocity by decreasing stride length and increasing slipping (reduce stride success ratio). At the smallest crevice height (4 mm), an increase in ceiling friction significantly decreased mean velocity [P < 0.001, F(2,72) = 37.87; Fig. 3B]. The decrease in velocity can be explained by a decrease in stride length [P < 0.001, F(2,72) = 122.58] and an increase in foot slippage [decrease in stride success ratio; P < 0.001, F(2,72) = 48.16]. Stride period remained unaffected by ceiling friction [P = 0.1, F(2,72) = 7.27]. For an increase in ground friction, we hypothesized that the trend in these locomotor variables would depend on the differential effect of ventral body drag versus an increased thrust from greater leg purchase. At the smallest crevice height (4 mm), an increase in ground friction significantly increased mean velocity at medium friction levels [P < 0.001, F(2,69) = 73.4] compared with both the low and high conditions (Fig. 3C). Stride length [P < 0.001, F(2,69) = 246.7] and stride success ratio [foot slippage; P < 0.001, F(2,69) = 89.10] showed a similarly significant trend with a performance peak at the medium friction condition. Stride period remained unaffected by changes in ground friction [P = 0.07, F(2,69) = 12.36]. Results from an increase in ground friction imply a trade-off between increasing leg thrust and increasing body drag. Therefore, confined-space crawling in cockroaches appears to have features of both the body friction of undulatory swimming in frictional fluids as seen in sand-swimming lizards (17, 43) and the thrust produced by flipper-driven surface locomotion produced by sea turtle hatchlings (8) except that in crevices the surrounding media do not flow (Fig. 3A). Body-friction legged crawling deserves further attention.

Fig. 3.
Fig. 3.

Effects of varying ground and ceiling friction for confined-space crawling performance. (A) Confined-space body-friction legged crawling characterized by drag on the dorsal and ventral surface of the body and friction-dominated thrust by legs in a nonflowing medium. (B) Performance metrics, velocity (grays), stride length (red-brown), and stride success ratio (blues) at 4-mm ceiling height (respective units indicated in parentheses below label) as a function of ceiling kinetic friction varied at three levels (low, medium, and high) with ground kinetic friction constant. (C) Performance metrics, velocity (grays), stride length (red-brown), and stride success ratio (blues) at 4-mm ceiling height (respective units indicated in parentheses) as a function of ground kinetic friction varied at three levels (low, medium, and high) with ceiling kinetic friction constant. Bars show mean ± 1 SD. Red stars represent a significant difference from other kinetic friction levels.

View this table:
Table S2.

Friction coefficients for dorsal and ventral surfaces relative to the body and feet

Model of Body-Friction Legged Crawling.

To begin to explain our experimental results and to generate further hypotheses concerning the effect of friction on confined-space crawling, we made a model with few fitting parameters (Fig. 4A; see Supporting Information for detailed description). The model consists of a compliant body confined between two parallel plates and actuated by a single leg. All animal–environment interactions are characterized by normal and tangential forces governed by Amonton’s and Coulomb’s laws of dry friction (51). Due to the lack of direct leg force measurements during confined crawling, we chose to model the cockroach leg as a linear force–velocity actuator (ref. 52, equation 1; Supporting Information, Eq. S13) motivated by studies of leg press tasks in humans (52), where the leg force and velocity can be approximated to vary linearly. The motion of the body is separated into two distinct phases—“stick” and “slip.” During stick phase (Supporting Information, Eqs. S8 and S9), maximum foot–ground friction is greater than the magnitude of the horizontal component of leg force, so the foot holds in place. As a result, the leg actuator displacement directly yields body displacement (Supporting Information, Eq. S10) and velocity (Supporting Information, Eq. S17). However, during slip phase, foot–ground friction is insufficient to prevent the foot from slipping backward relative to the body (Supporting Information, Eqs. S11 and S12). Body displacement and velocities are therefore computed by integrating net force (difference between thrust and drag) over the slip duration. The extent of leg excursions in stick and slip mode detailed in Supporting Information are illustrated in Fig. 4A. We used our measurements of friction coefficients (Table S2), normal (compressive) forces (Fig. 5B), maximum leg length (30 mm), and velocity (0.5 m⋅s−1), to predict stride length and mean body velocity for comparison with our experimental results (Figs. 3 B and C, and 4 B and C). We estimated maximum leg force from data on a related cockroach species (53). Computed body displacements during the stick phase were significantly longer than the slip phase at small crevice heights or high friction conditions, matching our experimental observations.

Fig. 4.
Fig. 4.

Model of body-friction legged crawling. (A, Left) Model simplified representation of a cockroach in a confined space depicted as a compressible body (light gray solid oval) with a single leg (wavy tan line) ending in a foot (dark gray box) confined within two parallel plates (hashed boxes). (Center) Free body diagram of foot and body. Leg force (FL) is indicated in tan, thrust (T) in blue, drag—ceiling (fc) and ground (fg)—in red, body weight (W), and all normal forces—ceiling–body (Nc), ground–body (Ng), and ground–foot (Nf)—in black. (Right) Foot positions, where forward body displacement occurs with (slip zone) and without (stick zone) foot slippage, are marked in green, whereas positions where no body motion is possible are indicated in red. Leg orientations at the above-foot positions—slip angle (transition from stick to slip, θslip) and the maximum (θmax) and minimum (θmin) angle to overcome body drag are indicated. (B) Performance metrics, velocity (gray) and stride length (red) at three ceiling heights (4.4, 4.7, and 5 mm) as a function of ceiling kinetic friction with ground kinetic friction constant. (C) Performance metrics, velocity (gray), and stride length (red) at three ceiling heights (4.4, 4.7, and 5 mm) as a function of ground kinetic friction with ceiling kinetic friction constant.

Fig. 5.
Fig. 5.

Material properties of cockroaches during compression and bioinspired robot. (A) Materials testing apparatus with custom-built chamber positioned atop load cell to measure force during cyclic compression (Movie S3). (B) Normalized body compressive force (measured force/body weight) as a function of crevice size (red) and abdomen strain (change in abdomen compression/maximum abdomen thickness). Abdominal strain increases from left to right corresponding to a decrease in crevice size. Blue lines show a compression rate of 0.5 mm⋅s−1. Tan lines show rate of 4 mm⋅s−1. Shaded bands represent 95% confidence limits. (Inset) Compressive force cycles as a function of body compression distance for two rates of compression. The corresponding crevice or ceiling height is shown for comparison. Areas within the loop represent the energy lost per cycle. (C) Prototype robot with adjustable sprawl and abdominal compression-inspired exoskeletal plate-like shell (Movie S4) Top row photos, Side view of freestanding and confined-space posture for robot between two surfaces with ceiling labeled. Bottom row photos, Front view of freestanding and confined-space posture for robot between two surfaces. Top black small arrow shows direction of compression, and bottom black arrows show leg sprawl direction when compressed. (D) Schematic of the robot with a bioinspired compressible shell to visualize the degrees of freedom in unconfined standing (Left) and confined sprawled (Right) postures. The robot uses inspiration from the cockroach, not only with respect to body compression but also by changes in leg posture, allowing effective contact by the foot (tarsus) when standing and the leg (tibia) when running in a confined space.

In accordance with our experimental data (Fig. 3B), the model indicates that increasing ceiling friction monotonically increases the overall resistance due to friction-based drag and, therefore, reduces forward velocity by decreasing stride length (Fig. 4B). The model suggests that smaller crevice heights are likely to decrease performance at a given ceiling friction. Furthermore, the model indicates that maximal forward velocity and stride length occur at intermediate ground friction conditions, because the magnitude of thrust depends on the differential friction between the foot and body (Fig. 4C). As in ceiling friction results, the model suggests that smaller crevice heights are likely to decrease performance at a given value of ground friction.

Although results from our initial model follow trends observed in experimental measurements, the model reveals at least three areas of future study for body-friction legged crawling. First, an improved understanding of foot contact mechanics is needed to provide insight into the complexities of thrust generation beyond simple coulomb friction-type attachments that include interlocking (14, 54, 55), adhesion (56⇓–58), and resistive forces in granular media (7, 59). Second, a leg actuation model that more effectively captures dependence of force production on such factors such as leg morphology, lever and transmission systems, and muscle mechanics (21). Finally, a more developed quantification of body and leg kinetic friction as they relate to contact geometry and exoskeletal material properties could improve estimates and reveal principles underlying the interesting complexity of the stick–slip interactions as a novel locomotor challenge for both animals (and robots). Incorporating these biologically relevant measurements into our initial model will move us closer to generating predictions concerning the habitats and environments animals might exploit using body-friction legged crawling.

Dynamic Compressive Forces.

As a first step toward quantifying the body’s exoskeletal material properties that enable cockroaches to traverse crevices and crawl in confined spaces, we performed a series of dynamic compressive cycle tests on living animals (Fig. 5A and SI Methods). We hypothesized that compression of the body and legs would demonstrate nonlinear, viscoelastic behavior, suggesting that crevice crossing might be affected by rate and that the magnitude of peak compression forces would reveal the extent of exoskeletal robustness. Compressive force showed a nonlinear increase as compression increased (Fig. 5B and Movie S3). Maximum average stresses ranged from 3.74 ± 0.56 kPa (at 0.5 mm⋅s−1) to 16.21 ± 3.20 kPa (at 4 mm⋅s−1) and were associated with very large strains (up to 0.50). Tangent modulus in the linear strain region (0.475–0.50) ranged from 0.11 ± 0.02 MPa (at 0.5 mm⋅s−1) to 0.50 ± 0.10 MPa (at 4 mm⋅s−1). The cockroach body behaved like a viscoelastic material with a resilience (percentage of energy return) of 60 ± 12% (at 0.5 mm⋅s−1) and 44 ± 10% (at 4 mm⋅s−1). At the smallest crevice or ceiling height of 3.2 mm, the net normal force (2.25 ± 1.17 N) on cockroaches averaged ∼300 times body weight (Fig. 5B). This is significantly higher than forces recorded in soft-bodied annelids that experience about 10 times body weight when compressed to 0.6 body diameter (60), indicating the exceptional capability of the exoskeleton. At the faster rate of compression, net forces (6.08 ± 1.61 N) attained values of 700 times body weight with a maximum of ∼900 times. We observed no damage to any body part. After the tests, cockroaches were able to fly normally and showed no significant changes in unconstrained running velocity (76.67 ± 7.34 cm⋅s−1) compared with controls [P = 0.49, F(1,35) = 6.21]. The relatively lower strain and higher buckling stiffness (3.36 GPa) measured in fresh sclerotized locust tibia (61) suggest the cockroach’s compressibility derives from their soft arthrodial membranes rather than compression or bending of stiff exoskeletal plates (41) or tubes (36).

Soft Robot with Legs.

Our discoveries from cockroaches inspired the design of a soft, legged hexapod robot named “compressible robot with articulated mechanisms” (CRAM) that we built using the SCM manufacturing approach (44⇓–46) involving laser-cutting, laminating, and the folding of exoskeletal-like plates (Fig. 5 C and D, and SI Methods). Like the animal, the robot successfully locomotes in vertically confined spaces by compressing its body in half (54%; 75–35 mm) and benefits from possessing a low friction shell (Movie S4). Although the fluid-filled hemocoel in the animal imposes constraints on how the body deforms (i.e., compression along one axis forces expansion along the remaining axis by fixed proportions governed by the bulk modulus), the robot is free of such restrictions allowing it to expand, for example, along the horizontal plane when confined vertically. The robot shell can withstand compressive forces (1-kg weight) 20 times body mass by benefitting from overlapping abdominal plates and dissipate impacts and collisions demonstrating capabilities of robots constructed using soft materials (62⇓–64). This prototype can increase sprawl angle (50° to 81°) by using compliant exoskeletal flexures to conform to the confined environment. Inspired by the cockroach, the terminal foot or tarsal-like segment is used for unconstrained running, whereas sprawling positions the leg or tibial-like segment for effective contact during compressed crawling (Movie S4). The advantages of dedicated propulsive appendages in compliant robots are clearly demonstrated by nearly an order of magnitude faster forward sustained speeds (both absolute, 14 cm⋅s−1, and relative to body length, 0.75 bl⋅s−1) compared with current state-of-the-art robotic crawlers (28, 63, 65, 66) despite large body compression. Future directions include addition of feet-like attachment mechanisms onto an upgraded leg design to improve peak forward velocity, control of leg actuation to improve energy efficiency (67), and the demonstration of new capabilities like turning (66), climbing (68), and jumping (64).

As Rus and Tolley (69) pointed out, “many of the exciting applications for soft robotics (such as search-and-rescue operations or environmental monitoring) require an autonomous, mobile system.” A major limitation for current soft robots is that they rely on power and/or control signals delivered through pneumatic (28) and/or electric tethers (65, 66), making them very heavy [for example, 1.2 kg (63)]. By relying on SCM manufacturing techniques that have proved highly successful in the design of a fast running hexapedal family of robots (70, 71), we have been able to make our palm-sized, confined-space crawling robot completely autonomous in power and control, while weighing just 46 g including onboard electronics and battery. A promising future direction for soft-arthropod–inspired legged robots is to combine the advantages of soft-bodied robots (26, 69) with appendages shown to be effective in tubular environments such as gastrointestinal tracts (68). Our bioinspired soft-robot prototype presented here is a first step toward a search-and-rescue robot that can perform crevice traversal and confined-space body-friction legged crawling in an effort to rapidly locate survivors.

Methods

Crevice Traversal.

We designed a custom apparatus to quantify the traversal behavior and time. The chamber consisted of clear acrylic plates with a vertically adjustable gate at one end (Fig. S1A). The inside of the entrance chamber was lined with 40-grit sandpaper to ensure effective footholds. Based on exoskeletal body compression measurements, we set the gate height at 3.2, 4.4, and 6.1 mm. We collected 315 trials from four individuals, classified them as (i) success, (ii) failure–turn back, or (iii) failure–stuck, and analyzed the 100 successful traversals (Fig. 1).

Confined-Space Crawling.

We built a second custom apparatus for confined-space crawling that consisted of a closed, horizontal clear acrylic track with a vertically adjustable ceiling height (4–12 mm; Fig. S1B). For the first set of kinematic experiments consisting of 104 trials of five individuals each tested at 4, 6, 9, and 12 mm, we lined the inside bottom surface of the track with 40-grit sandpaper, whereas the inside top surface was acrylic. In a second set of experiments consisting of 638 trials on 10 individuals, we varied ceiling and ground friction based on our direct measurements for four ceiling heights. We present the 4-mm ceiling height data for 155 trials on 10 individuals (Fig. 3). In one condition, we varied ceiling kinetic friction from 0.09 to 0.46, while the ground kinetic friction was kept constant at 1.73. In the other condition, we varied ground kinetic friction on the body from 0.37 to 0.76 and from 0.84 to 2.04 on the legs (tibial spines), while the ceiling kinetic friction was kept constant at 0.16. We measured velocity, stride length and period, and stride success ratio. Based on our results, we developed a model of body-friction legged crawling (Fig. 4 and Supporting Information).

Dynamic Compressive Forces.

We constructed a clear acrylic cylindrical chamber with the top attached to a freely sliding ceiling and the bottom mounted directly onto a universal materials testing machine load cell (Fig. 5A). We tested five live animals 10 times at two compression speeds, 0.5 and 4 mm⋅s−1, over a range of compression values matching those experienced by animals in crevice traversal and confined-space crawling. We constructed force–compression curves, noted maximum net force, determined stiffness, and calculated resilience.

Soft Robot with Legs.

We built a robot that uses a combination of postural adjustment from leg reorientation and body compression to locomote in vertically confined spaces, based on the cockroach data. The key design innovation making this possible is a flexible back spine coupled with a deformable shell made of overlapping plates similar to the exoskeletal plates of a cockroach abdomen (Fig. 5 C and D, Movie S4, and Supporting Information). The design of the robot is realized using the SCM manufacturing technique (44⇓–46).

SI Methods

Animals.

We used a total of 21 male cockroaches (Periplaneta americana; Carolina Biological Supply) distributed among our six experiments with an average mass of 0.76 ± 0.16 g (mean ± SD) and body length of 31.10 ± 2.16 mm. Before experimentation, cockroaches were kept in communal plastic containers at room temperature (22 °C) on a 12-h/12-h light/dark cycle and provided water and food (fruit and dog chow) ad libitum.

Animal Experimental Design and Protocol.

All experiments were performed at 28 ± 2 °C (mean ± SD). Before starting any experiment, each cockroach was marked with retro-reflective paint on the dorsal surface—two on the pronotum, one at the base of the abdomen, one at the tip of tibia–tarsus joint of each hind leg to aid tracking. The dorsal body markers were used to calculate crawling velocity and tortuosity, whereas the leg markers were used to estimate gait, tarsus midline distance, stride success ratio, and stride length and period. We attained a balanced experimental design by running the same animal across each of the different conditions for a minimum of five (and a maximum of eight) trials. Thus, each individual served as its own control, allowing us to use paired statistics.

Kinematics.

We recorded movies of cockroaches at 500 frames⋅s−1 with a resolution of 1,280 × 1,024 pixels using synchronized high-speed movie cameras. One camera was positioned directly above the track, capturing the top view, and the other recorded the side view. Images were buffered through the camera memory until posttriggering, after which a movie clip (4 s) from each camera was reviewed and cropped to the relevant time segment. All movie capture, downloading, and conversion were done with a proprietary software program. We determined the kinematics of the behavior from the movies using custom motion tracking software.

Statistics.

We used repeated-measures ANOVA and Cochran–Mantel–Haenszel tests for continuous and nominal variables, respectively, with Tukey’s honest significant difference for post hoc contrast analyses in JMP software (SAS, Inc.). A repeated-measures design with a mixed model was used to determine the effect of condition. In our model, the condition (crevice size/ceiling height as the case may be) was included as a fixed effect while the animal was included as random effect. The response was our performance metric (running velocity, stride period, stride length, etc.). We used a standard least-squares personality with reduced maximum likelihood as our method to fit our data. We report the P value, F ratio in American Psychological Association format to support/reject our hypothesis as appropriate. For categorical responses (success), we report the χ2 value. We used an α level of 0.05 for all statistical tests.

Exoskeletal Compression.

We directly measured the sagittal plane deformation of the cockroach exoskeleton for freshly anesthetized animals at 10%, 25%, 37%, 50%, and 75% along the body from head to cerci upon a flat ground surface (Fig. 1C, Table S1, and Fig. S1C). The lengths along the body correspond to the head–pronotum junction, front leg, middle leg, and hind leg coxa sections, and middle of the abdomen, respectively. To determine body compressibility, we obtained sagittal view measurements under two conditions—unloaded with an intact, undisturbed animal and loaded by suspending a 100-g load with a light string across the body segment at the given length. Five sets of loading and unloading cycles were performed at each of the five cross-sections.

Crevice Traversal.

To measure horizontal crevice traversal performance, we constructed a chamber consisting of clear acrylic tube (square cross-section, 2.54 × 2.54 cm) with an opening at one end and a vertically adjustable gate also made of clear acrylic at the other (Fig. S1A). The gate was mounted on a precision miniature linear stage (model 422-1s; Newport Corporation) that allowed fine control (±0.05 mm) in the range of 2–10 mm. The insides of the acrylic chamber, with the exception of movie recording ports on the front and top, were lined with 40-grit sandpaper to ensure effective footholds. Our exoskeletal compression measurements led us to test performance at three different crevice heights—3.2, 4.4, and 6.1 mm.

To encourage the animals to traverse the horizontal crevice, we evoked an escape response by light stimulation of their cerci or by gently blowing until they entered the apparatus. The open end was then sealed off, leaving the crevice under the gate as the only escape outlet. In addition, we shined bright light from the sealed direction, dissuading the animals from turning around. Animals typically searched and explored the opening using their antenna (48) and then rushed through. Successful trials were operationally defined as those where the animals were able to pass through the gate completely without stopping as determined by their cycling of the legs. Failures included both turning back away from the crevice as well as getting their body stuck or wedged within the crevice (Fig. 1) in the duration of the trial (about 5 min).

Confined-Space Crawling.

To quantify confined-space crawling, we constructed a closed horizontal clear acrylic chamber, 25 cm long and 10 cm wide, with a vertically adjustable ceiling (Fig. S1B). The ceiling was mounted on precision miniature linear stages (model 422-1s; Newport Corporation) that allowed fine control (±0.05 mm) in the range of 4–12 mm.

To encourage the animals to crawl rapidly through the confined space, we evoked an escape response by light stimulation of their cerci or by gently blowing. We accepted trials when the animal crawled within the confined space for a minimum of 15 cm. We rejected trials where (i) the cockroaches stopped or tried to climb the side wall, (ii) their body (excluding their legs) collided with the side wall, or (iii) they exhibited turns of more than 150°.

Ceiling Height Variation.

We tested animals under four different ceiling heights—4, 6, 9, and 12 mm, respectively (Fig. 2A), and used 40-grit sandpaper for the track bottom surface and acrylic for the inside top surface (Fig. S1B). We selected these particular sizes for the following reasons:

  • 12 mm (Free running height): The typical height of the dorsal surface of the cockroaches during unconstrained running. We used this as the control condition.

  • 9 mm (Crouched height): The maximum height that the animals could locomote without making continuous contact with the ceiling. They adopted a suitable crouched posture by adjusting their leg joint angles and lowered their center of mass.

  • 6 mm (Body compression height): The mean height of the thickest abdominal section measured without compression. The height represented the smallest size before which the animal was forced to necessarily deform its body in compression to successfully negotiate the confined space.

  • 4 mm (Minimum ceiling height): The smallest ceiling height that cockroaches could successfully complete at least five trials based on our operational definition.

To evaluate confined-space crawling performance, we calculated velocity, stride length, period, sprawl angle, tarsus centerline distance, stride success ratio, and tortuosity (Fig. 2). We defined sprawl angle as the average angle of the vector from the tip of the tarsus when in contact with the ground to the cockroach’s center of mass relative to the sagittal plane. We calculated the tarsus midline distance as the perpendicular distance of the metathoracic leg tarsus–ground contact point to the midline of the cockroach body approximated as the line joining the midpoint of the two pronotum markers with the one at the base of the abdomen. We defined stride success ratio as the ratio of successful strides (with no foot slipping) to the total number of strides. We determined tortuosity index to be the ratio of the forward displacement of the cockroach in the chamber relative to the length of the actual path taken.

Surface Friction Variation.

We tested the effect of varying contact surface friction on cockroach performance across the four ceiling heights. We varied ceiling and ground kinetic friction with respect to the animal independently across three kinetic friction levels in two separate sets of experiments. For variation in ceiling kinetic friction, we used graphite powder over acrylic, uncoated acrylic and 40-grit sandpaper as dorsal contact surface for a low, medium, and high condition, respectively, whereas 40-grit sandpaper served as the ground contact surface. For variation in ground kinetic friction, we used 100-grit, 40-grit, and 20-grit sandpaper as ground contact surface for a low, intermediate, and high conditions, respectively, whereas uncoated acrylic served as the ceiling contact surface.

To quantify friction on body (drag), freshly anesthetized animals were mounted onto a six-axis force-torque sensor (Nano17; ATI, Inc.). A flat plate mounted on a linear stage (model 422-1s; Newport Corporation) at 4-mm vertical offset to the sensor top was coated with the test material and dragged along horizontally at a constant speed that matched the animal’s mean running speed. Data were recorded at 1,000 Hz. We noted the mean normal force, maximum (peak drag) and settling value (sliding drag) of the horizontal force for each trial. The ratios of the peak drag and sliding drag to the mean normal force were defined as the coefficients of static and kinetic friction, respectively. Compressed animals appeared to be using their tibia spines for thrust, so to determine the friction coefficient with feet (lower bound) on the ground surface, we ablated animal’s tarsus, ran the animals on a semicircular track of radius 60 cm, and measured the critical angle of failure (falling off or sliding down the track), and quantified the tangent of failure angle as leg friction coefficient (Table S2).

We used the aforementioned friction measurements to set a range for kinetic friction. Assuming that our cockroaches have force production capabilities similar to that measured in a related species (53), we estimate the maximum force produced by each leg to be ∼38.0 mN (5.1 times mean body weight of 0.76 g). This suggests that maximum forward thrust estimate (227.9 mN) can overcome the estimated body drag (24.71–224.80 mN) and likely pushes the animal to near-maximal performance [comparable to burrowing performance in razor clams (25), earthworms (60), and ocellated skinks (72)].

Dynamic Compressive Forces.

To measure the material properties using a universal testing machine (Instron Industrial Products; 5900 series), we constructed a clear acrylic cylindrical chamber of diameter 5.08 cm with a freely sliding ceiling (Fig. 5A). The chamber was mounted directly onto the lower fixture of the machine, whereas the ceiling was mounted onto the movable fixture comprising the load cell. A mechanical stop was added at 2-mm chamber height to prevent fatal damage to the animal.

We placed a live animal in the custom chamber with the initial ceiling height set to 15 mm. Using a triangular wave profile set, we performed cyclic compression tests on the animal by moving the ceiling between 15 and 3 mm at two speeds, 0.5 and 4 mm⋅s−1, over a range of compression values matching what animals experienced in crevice traversal and confined-space crawling. The typical cockroach body area over which the compression load is distributed is about 3 cm2. We measured force–compression curves, recorded maximum net force, estimated stiffness as a function of compression distance and calculated resilience as the percentage of energy returned during relaxation to the energy input during compression. Before and after performing the test, we measured each animal’s average straight-line speed over 25 cm and their ability to fly.

Robot Design—Compressible Robot with Articulated Mechanisms (CRAM).

Using a combination of postural adjustment from leg reorientation and body compression, cockroaches traversed and locomoted in vertically confined spaces. With this inspiration, we designed a robot that uses the above two principles to passively adjust its sprawl angle by reducing its vertical height and thus conforms to the confined space. The key to robot performance is a flexible back spine (along the longitudinal axis) and a deformable shell made of overlapping plates similar to the exoskeletal plates of a cockroach abdomen (Fig. 5 C and D, and Movie S4). The design of the robot is realized using the smart composite microstructures (SCM) manufacturing technique. The rigid elements of the robot are made of a poster board (4-ply Railroad Board; Peacock, Inc.) composite laminate with 0.001 in polyester sheet (Dura-lar; Grafix, Inc.) serving as the flexure layer. To construct the low-friction, deformable shell, we used 0.005-inch polyester sheet (Dura-lar; Grafix) as the stiffer material. The legs of the robot are L-shaped to allow good ground contact in the standing and maximally sprawled configurations and is 3D printed (ProJet3500; 3DSystems, Inc.). The drive mechanism comprises two miniature DC motors (MK-07; Didel, Inc.) with custom 3D printed gearing operating independently to control the left and right side of the robot. We used commercially available electronics (Dashboard; Dash Robotics Inc.) to control the robot dynamics in open loop over Bluetooth 4.0. The resulting robot prototype is small [18 cm long, 75 mm high (unrestricted)], lightweight (46 g, with onboard control electronics and battery), and capable of crawling in vertical confinements as small as 35 mm in height.

The robot’s maximum percentage vertical compression was 54% (75–35 mm). Maximum load-bearing capacity of exoskeleton was ∼1,000 g, greater than 20 times body mass. The robot’s sprawl angle when unconfined was 50°, whereas in the unconfined at 35 mm it was as great as 81°. The robot’s unconfined speed at 75-mm ceiling height was ∼27 cm⋅s−1 (1.5 bl⋅s−1), whereas its confined speed compressed at 35 mm was 14 cm⋅s−1 (0.75 bl⋅s−1). The robot’s maximum stride frequency was 25 Hz, with a mean stride length unconfined at 75 mm of 1.1 cm and confined at 35 mm of ∼0.6 cm.

SI Results and Discussion

Single-Leg Compliant Body Model for Confined-Space Body-Friction Legged Crawling.

To begin to understand the mechanics of cockroaches crawling in confined spaces, let us consider the following simplified model. A cockroach is reduced to a three-link model comprising a compressible body confined between two flat plates a distance 2h apart, a single leg (leg length, l) hinged at the body center of mass (pin joint) and terminating in a foot (Fig. 4A). The body slides along the top and bottom surfaces of the confined space at all times. The foot, whose mass is insignificant compared with that of the body, makes contact only with the ground surface. We simplify the complex interactions (adhesion, interlocking, etc.) of the foot structures with the ground surface into one governed by friction. Thus, all surface interactions (including foot–ground) are modeled to be Coulomb friction based, i.e., the tangential surface forces are proportional to normal forces (proportionality constant is coefficient of friction), and independent of speed. We further consider that coefficients of static and kinetic friction to be identical.

To begin to quantify the dynamics, we generate free body diagrams at the foot and body (Fig. 4A). Balancing forces at the foot at leg orientation, θ, yields the following relationships:Nf=FL⁡sin(θ),[S1]ff=FL⁡cos(θ),[S2]ff_max=μfFL⁡sin(θ).[S3]As the leg starts cycling, the friction force at the foot (ff) until its maximum (ff_max), balances the horizontal component of the leg force (FL) and prevents the foot from slipping. We refer to this condition as the “stick” regime because the foot sticks to the ground surface (no relative movement). Under this condition, the leg extension is equal to the body displacement and, therefore, can be determined directly from the leg kinematics. As the leg angle (θ) decreases, the horizontal (or forward) component increases until it reaches max friction from the surface (ff_max). We refer to this critical angle as the “slip angle” (θslip) and for all leg angles smaller than slip angle, the foot slips (and hence referred to as the “slip” regime; Fig. 4B) and hence the body displacement is lesser than the leg displacement and needs to be computed using kinetics (or inertial methods using force balance):θslip=cot−1(μf).[S4]At the end of slip phase, we assume that the leg is instantaneously returned to touchdown position (perpendicular to the surface). To compute body displacements, we need to first determine the limits of leg extension—where the thrust (T) from the legs overcomes the frictional drag (D) on the body. The same can be obtained via force balance on the body:T=min{FL⁡cos(θ),μfFL⁡sin(θ)},[S5]T≥D=fc+fg= μcNc+μgNg,[S6]Nc+mg=Ng+FL⁡sin(θ).[S7]From our experimental measurements (Table S2) and previous literature (53), we find that the animal body weight is at least an order of magnitude smaller than the other forces involved and hence is assumed to be insignificant. To determine the leg excursion during stick mode, we estimate the maximum leg angle (θmax) until which thrust overcomes drag (Fig. 4B):T=Fmax⁡cos(θmax);D=(μc+μg)Ng+μcFmax⁡sin(θmax);[S8]LetCh= NgFmax,θmax= cot−1(μc)−sin−1(Ch(μc+μg)√(1+μc2)).[S9]Therefore, the body displacement under stick mode per leg cycle (xstick) is determined as follows:xstick=h(cot(θslip)−cot(θmax))2.[S10]Using a similar approach, we can calculate the extent of slip mode by estimating the minimum leg angle (θmin) when thrust overcomes drag, provided it is greater than the kinematic limit [cot−1(h/2L); Fig. 4B]:T=μfFmax⁡sin(θmin);D=(μc+μg)Ng+μcFmax⁡sin(θmin);[S11]θmin= sin−1(Ch(μc+μg)(μf−μc))≥cot−1(h2L).[S12]To determine body displacement under slip mode, we need to know the velocity at the end of stick mode and use Newton’s second law thereafter. Therefore, we need an actuator model for the leg. Let us consider a simple, reciprocating leg actuator with a linearly decreasing force–velocity relationship (52) limited in both force (Fmax) and velocity (vmax) (Eq. S13) driven by a controller that attempts to output maximum propulsive force as possible during slip phase. Furthermore, we assume that the leg forces are sufficient to overcome drag (at least at some leg orientation) but insufficient to compress the body (thus unable to reduce normal forces due to confinement):FLFmax+vLvmax=1.[S13]Balancing forces on the body in the horizontal direction during stick mode,FL⁡cos(θ)=(μc+μg)Ng+μcFL⁡sin(θ),FLFmax=Ch(μc+μg)cos(θ)−μc⁡sin(θ),vLvmax=1−Ch(μc+μg)cos(θ)−μc⁡sin(θ).[S14]Furthermore, the body velocity (vb) and leg velocity (vL) can be related using the geometric constraint below:x2+ h2=l2Differentiating,xx˙= ll˙→ x˙=(lx)l˙vb=vL⁡sec(θ).[S15]Combining Eqs. S14 and S15, we obtain the velocity of the body under stick (vstick) as follows:vstickvmax=(1−Ch(μc+μg)cos(θ)−μc⁡sin(θ))sec(θ).[S16]The mean velocity (v˜stick) during stick mode is obtained by the following equation and is computed numerically:v˜stick=vmax∫θslipθmax(1−Ch(μc+μg)cos(θ)−μc⁡sin(θ))sec(θ)dθ∫θslipθmaxdθ.[S17]Additionally, the stick duration (tstick) is as follows:tstick=xstickv˜stick.[S18]The velocity at end of stick mode (or beginning of slip mode, v0) is obtained as follows:v0=vmax(1−Ch(μc+μg)cos(θslip)−μc⁡sin(θslip))sec(θslip).[S19]Assuming that the leg slides back at the fastest speed it can once the foot starts slipping, we can estimate the net slip duration (τslip):τslip= h(cot(θmin)−cot(θslip))2vmax∫θminθslip⁡sec(θ)dθ∫θminθslipdθ.[S20]We can compute the instantaneous and mean acceleration (aslip and a˜slip, respectively) on the body during the slip mode as follows:maslip=(μf−μc)FL⁡sin(θ)−(μc+μg)Ng,a˜slip=∫θminθslip((μf−μc)FL⁡sin(θ)−(μc+μg)Ng)dθ∫θminθslipdθ.[S21]Therefore, the instantaneous and mean body velocities (vslip and v˜slip, respectively) are computed as below:vslip=v0−a˜slipt,t≤τslip,tslip=v0a˜slip,tslip≤τslip,v˜slip= v0−a˜sliptslip2.[S22]The step length, i.e., body displacement during this period is computed as below:xslip=v˜sliptslip.[S23]Therefore, the total step length (xstep) is as follows:xstep=xstick+xslip.[S24]Therefore, the mean step velocity (vstep) is as follows:v˜step=xsteptstick+tslip.[S25]Finally, assuming a stride to be two steps like in a hexapedal gait observed in cockroaches, we estimated the performance metrics, stride length, and mean forward velocity, as follows:v˜forward=v˜step,[S26]xstride=2xstep.[S27]The computed body displacement in the slip regime was considerably smaller relative to that in the stick regime and did not affect the overall trends. Using the above formulation, we quantified (Fig. 4B) the effect of friction (ceiling, ground) and gap size on performance (stride length, forward velocity).

List of Symbols Used.

  • Nf, Normal force at foot from ground;

  • Nc, Normal force on body from ceiling;

  • Ng, Normal force on body from ground;

  • μf, Friction coefficient between foot and ground;

  • μc, Friction coefficient between body and ceiling;

  • μg, Friction coefficient between body and ground;

  • ff, Friction force at foot;

  • ff_max, Maximum possible friction force at foot;

  • fc, Friction force on the body from ceiling;

  • fg, Friction force on the body from ground;

  • T, Thrust on the body;

  • D, Drag on the body;

  • FL, Leg force;

  • Fmax, Maximum leg force;

  • Ch, Normalized compression;

  • θ, Instantaneous leg angle with ground;

  • θslip, Slip angle;

  • θmax, Maximum leg angle to overcome body drag;

  • θmin, Minimum leg angle to overcome body drag;

  • h, Crevice/confined-space ceiling height;

  • L, Maximum leg length;

  • l, Instantaneous leg length;

  • x, Instantaneous step length;

  • xstick, Step length under stick mode;

  • xslip, Step length under slip mode;

  • xstride, Stride length;

  • vstick, Instantaneous velocity under stick mode;

  • v˜stick, Mean velocity under stick mode;

  • vslip, Instantaneous velocity under slip mode;

  • v˜slip, Mean velocity under slip mode;

  • vmax, Maximum leg velocity;

  • vb, Instantaneous body velocity;

  • v0, Velocity at end of stick or start of slip mode;

  • v˜forward, Mean body forward velocity;

  • aslip, Instantaneous acceleration under slip mode;

  • ãslip, Mean acceleration under slip mode;

  • t, Instantaneous time;

  • tslip, Step period under slip mode;

  • τslip, Net slip duration.

Acknowledgments

We thank Dan Goldman for modeling advice, Pauline Jennings for movie editing and figure production, Tom Libby for assisting with figure photos, and Anand Mohapatra and the 2012 class of IB135L Mechanics of Organisms Laboratory for preliminary data collection. We thank the Biomimetic Millisystems Laboratory of Ron Fearing at University of California, Berkeley, and DASH Robotics, Inc. We also thank Marvalee Wake for reading the manuscript. This work was supported by an ARL MAST CTA grant (to R.J.F.).

Footnotes

  • ↵1To whom correspondence should be addressed. Email: rjfull{at}berkeley.edu.

  • Author contributions: K.J. and R.J.F. designed research; K.J. performed research; K.J. and R.J.F. analyzed data; and K.J. and R.J.F. wrote the paper.

  • The authors declare no conflict of interest.

  • This article is a PNAS Direct Submission. Z.J.W. is a guest editor invited by the Editorial Board.

  • This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10.1073/pnas.1514591113/-/DCSupplemental.

Freely available online through the PNAS open access option.

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Cockroaches in crevices and confined spaces
Kaushik Jayaram, Robert J. Full
Proceedings of the National Academy of Sciences Feb 2016, 113 (8) E950-E957; DOI: 10.1073/pnas.1514591113
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