Terradynamically streamlined shapes in animals and robots enhance traversability through densely cluttered terrain

For animal experiments, we used five adult male discoid cockroaches, B. discoidalis (Mulberry Farms, Fallbrook, CA, USA), as females were often gravid and, therefore, under different load-bearing conditions. Prior to and during experimentation, we kept the cockroaches in communal plastic containers at room temperature (28 °C) on a 12 h: 12 h light: dark cycle and provided water and food (fruit and dog chow) ad libitum. The discoid cockroach has a relatively thin, rounded body, the dorsal surface of which resembles a slice of an ellipsoid (figure 2(A)). See table 1 for animal body mass and dimensions.

For robot experiments, we used VelociRoACH, a six-legged robot inspired by cockroaches (figure 2(B), Haldane et al 2013). The robot has springy c-legs and uses an alternating tripod gait to run at ∼30 body length s⁻¹ with dynamics described by the Spring-Loaded Inverted Pendulum (SLIP) Model (Blickhan 1989, Full and Tu 1990). However, unlike the discoid cockroach, the robot has a cuboidal body, with flat faces, straight edges, and sharp corners (figure 2(B)), resulting in high angularity and low roundness. We ran the robot with an open-loop control algorithm to test only the effect of mechanical interaction. See table 1 for robot body mass and dimensions.

We designed and constructed an apparatus to create a laboratory model of grass-like, cantilever beam obstacles that allowed precise control and systematic variation of the beam obstacles' geometric parameters and mechanical properties (figure 3(A)). The model beam track measured 180 cm long by 24 cm wide and was constructed using aluminum struts and acrylic (McMaster Carr, Elmhurst, IL, USA). The beams used in the track were manufactured by laser-cutting paper (VLS6.60, universal Laser Systems, Scottsdale, AZ, USA). Control and variation of beam width, thickness, lateral spacing was attained by modifying laser cutting patterns (Adobe Illustrator, Adobe Systems, San Jose, CA, USA). Control and variation of beam stiffness and damping was achieved by using different paper properties and modulation of beam dimensions. Control and variation of beam height, angle, and fore-aft spacing was attained by modification of the two parallel plates in which the beams were inserted and the two rails on which these two plates sat. For the animal experiments, we used thin cardstock (Wausau Exact Index Premium Cardstock, 199 g cm⁻², 279 × 216 mm², Wausau paper, Harrodsburg, KY, USA) to manufacture the beams. For the robot experiments, we used thick cardstock (Pacon 6-ply railroad board, Appleton, WI, USA). Mechanical tests showed that the model beams had a Young's modulus (∼10⁹ Pa) within the natural range that we found for small grass and thin plant stems (∼10⁸–10¹⁰ Pa) by direct measurement, and thus a similar stiffness due to their similar geometry.

In animal experiments, we challenged cockroaches to go through a beam obstacle field (figure 3(A)) consisting of three layers of vertical, tall beams taller than twice the animal body length. Lateral spacing was less than half the animal's body width, slightly less than the animal's standing height, but slightly larger than the animal's body thickness without legs. The fore-aft spacing between each adjacent layer was less than half the animal's body length (table 1).

In robot experiments, the beam obstacle field (figure 3(B)) consisted of three layers of vertical, tall beams. Beams were taller than twice the robot body length, with lateral spacing larger than the robot's body width, but smaller than the width of the artificial shells (12 cm, figure 3(D); see section 2.4). The fore-aft spacing between each adjacent layer was less than half the robot body length (table 1).

We first tested the animal with an unmodified body shape. To determine whether and how an animal's body shape affected its traversability through beam obstacles, we further tested the same individuals with modified body shapes by sequentially adding artificial shells to the animal's dorsal surface with the following shapes (figure 3(C), table 2): (1) an oval cone with similar rounded shape to the animal's body; (2) a flat oval to remove three-dimensional roundness; and (3) a flat rectangle to further reduce two-dimensional roundness. Finally, to control for long-term learning and fatigue, we tested unmodified animals again after removing the shells. In preliminary experiments, we randomized the order of presentation of the three shells and found no effect. By testing all shells on the same individuals, they served as their own control.

We used a small amount of hot glue to securely attach the artificial shells to the animal's hard pronotum. The shells were made of thick cardstock (Pacon 6-ply railroad board, Appleton, WI, USA). The maximal length and width of the three shells for each animal were the same as its body length and width. The shells represented a small increase in the animal's mass (table 2), but significantly increased the overall body volume and surface friction. The kinetic friction coefficient was 0.54 ± 0.03 (measured by the inclined plane method) between the shell and beams, greater than 0.10 ± 0.01 between the animal body and beams (P < 0.05, Student's t-test). The animal running on flat ground did not slow down with the shells (64 ± 11 cm s⁻¹) compared to without shells (67 ± 11 cm s⁻¹; P > 0.05, repeated-measures ANOVA).

From our hypothesis that the discoid cockroach's thin, rounded body facilitates its ability to traverse three-dimensional, multi-component terrain, we predicted that: (1) with unmodified body shape, the animal's obstacle traversal performance will be at its maximum; (2) with the oval cone shell, traversal performance will decrease due to increased volume and surface friction; and (3) with the flat oval and rectangle shells performance will be further decreased as roundness is reduced, and obstacle traversal pathways altered. Assuming negligible learning and fatigue, we expected the animal to recover its obstacle traversal pathways and performance after the shells were removed.

In robot experiments, we also modified the robot's body shape by adding exoskeletal shells of similar shapes to those in the animal experiments (figure 3(D)): an ellipsoidal shell (Haldane et al 2015), and a flat oval shell, and a flat rectangle shell (table 3). Although all the shells were comparable to the robot in mass, adding them did not slow down the robot on flat ground (60 cm s⁻¹ with or without shells at 10 Hz).

2.4.1. Experimental protocol

2.4.2. Data analysis and statistics

From the videos, we obtained the animal and robot locomotor performance metrics including traversal probability and time. We also obtained the locomotor ethogram for each trial, and performed a locomotor pathway ethogram analysis (Blaesing and Cruse 2004b, Harley et al 2009, Daltorio et al 2013) to quantify the locomotor pathways of traversal.

In animal experiments, we collected a total of 1050 trials from five individuals each with five shape treatments, with approximately 40 trials for each individual and shape combination. We rejected trials in which the animal spontaneously ran towards the beam obstacle field (rather than after being chased by the experimenter), because during spontaneous locomotion the animal often slowly explored and traversed the obstacle field rather than attempting to negotiate it as fast as it could during the escape response. Including trials with slow exploration would bias the traversal time results. We also rejected trials in which the animal used the sidewalls of the track to negotiate the beam obstacles. With these criteria, we accepted 869 trials from five individuals each with five shapes, approximately 35 trials for each individual and shape combination. This large sample size enabled us to obtain statistically meaningful data of the distribution and pathways of the diverse locomotor modes observed (see section 3.1).

To analyze obstacle traversal pathways and performance for each individual and shape treatment, we calculated the averages of all locomotor metrics (e.g., traversal probability, traversal time, transition probability between locomotor modes) using all the trials from the same individual with each shape treatment. To compare between each shape treatment, we then calculated the averages of each locomotor metric by averaging the means of all five individuals for each shape treatment. All average data are given in mean ± s.d., unless otherwise specified. We used repeated-measures ANOVA to test whether each locomotor metric (averages of the means of five individuals) differed among body shapes. We used Tukey's honestly significant difference test for post-hoc analysis where needed.

To begin to differentiate the effects of body shape versus active feedback resulting in altered leg thrusts, we pulled deceased animals through the vertical beam field (figure 4). A stepper motor (ROB-09238, SparkFun Electronics, Boulder, CO, USA) pulled freshly deceased animals through the beam obstacle field using a wire with an end glued to the dorsal surface of the animal's pronotum at a constant speed (10 cm s⁻¹), while videos were recorded from top, side, and rear views using three webcams (C920, Logitech, Newark, CA, USA). Videos were analyzed to determine the movement of the animal body during interaction with the beams. We added the same three shell shapes to the deceased animals. For each shape treatment, we performed ten pulling trials. At the beginning of each pulling trial, the animal body was placed at the same location, and care was taken to ensure that the wire was in the middle of two adjacent beams with minimal friction.

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In robot experiments, we collected 15 trials for each shape treatment, and used ANOVA for statistical testing.

We first focused on the animal's beam obstacle traversal with an unmodified body shape. The animal's obstacle traversal was composed of four phases: approaching and collision, exploration, negotiation, and departure. (I) Approaching and collision. The animal ran rapidly (67 ± 11 cm s⁻¹) towards the beams in a normal, horizontal body orientation, collided with the beams, and was forced to stop. This occurred for all the trials during the escape response. (II) Exploration. In approximately one third of all trials (34 ± 20%), before entering the obstacle field, the animal attempted to push through the beams (during which the body often pitched up slightly), then moved laterally (by at least one lateral beam offset, or 2 cm) to explore the first layer of beams in an attempt to find the boundary of the obstacles, during which it swept its antenna to detect openings. (III) Obstacle negotiation. The animal then negotiated the beam obstacles using one or a sequence of locomotor modes. (IV) Departure. In traversed trials, the animal ran away from the obstacle field and into the shelter, again, in a normal, horizontal body orientation.

During the obstacle negotiation phase, the animal displayed a diversity of distinct locomotor modes (figure 5):

• (A) Roll maneuver (red; movie S1 stacks.iop.org/BB/10/046003/mmedia): the animal rolled its body to either side (near 90°) such that its smallest body dimension (thickness) fit within the narrow lateral spacings between beams. In most cases after rolling, the animal then maintained its body orientation and maneuvered through the narrow gaps between the vertical beams by pushing its legs against the beams. In this mode, the legs adopted a more sprawled posture as compared to the running leg posture on flat ground and did not always use an alternating tripod gait. The pronotum joint also occasionally flexed and/or twisted slightly during the maneuver. Before the roll maneuver was initiated, the animal's body often briefly pitched up slightly as the animal pushed against the beams, but to a smaller degree than the body rolling. In some cases after rolling, the animal did not maneuver through the beams, but transitioned to other modes.
• (B) Pushing (magenta): the animal maintained a horizontal body orientation and pushed forward through the beam obstacles. In this mode, the animal's body was often elevated by the bent beams and its legs thus lost contact with the ground and pushed against the beams.
• (C) Lateral movement (orange): between the layers of beams, the animal changed its heading turning left or right and moved in the lateral (±y) direction.
• (D) Hiding (yellow): the animal stopped moving and hid within the space between the beams of the obstacle field, likely using the shadows of the beams as a shelter.
• (E) Returning (dark green): the animal returned (or occasionally stopped moving) without entering the beam obstacle field.
• (F) Climbing (light green): the animal pitched its body head-up, grabbed onto the beams with its legs, and climbed upward along the beams.
• (G) Falling forward (dark blue): after climbing up the beams, the animal fell to the ground across the beam obstacle field as the beams bent nearly 90° forward from the animal's weight.
• (H) Falling back (cyan): after climbing up the beams, the animal fell to the ground behind the beam obstacle field as the beams bent nearly 90° backward from the animal's weight.
• (I) Falling down (light blue): after climbing up the beams, the animal lost foothold and fell down to the ground within the beam obstacle field.

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Our locomotor pathway ethogram showed that movement of the animal through beam obstacles emerged via complex locomotor pathways (figure 6(A)). This differed from previous studies in insects and reptiles negotiating simpler obstacles like one or two steps, a gap, a ditch, or an incline (Watson et al 2002a, 2002b, Bernd et al 2002, Blaesing and Cruse 2004b, Ritzmann et al 2005, Kohlsdorf and Biewener 2006, Theunissen and Dürr 2013) in which only moderate changes in leg kinematics and body orientation were observed, and was most similar to cockroaches negotiating a shelf obstacle where more than one distinct locomotor mode (e.g., climbing and tunneling) was observed (Harley et al 2009). This also contrasted with locomotion on simpler, near two-dimensional ground where the observed movements (e.g. running, walking, hopping, climbing) were often stereotyped.

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Despite the considerable diversity of locomotor modes and pathways of traversal (figure 6(A)), we discovered that when body shape was unmodified, the animal most frequently (nearly half of the time) used roll maneuvers alone to traverse the beam obstacles (48 ± 9%; table 4; figure 6(A), red label; figure 8(A), unmodified before, red), without employing any other mode(s). Traversal time using roll maneuvers alone (2.0 ± 0.5 s; figure 6(B), red) was shorter than using any other locomotor mode(s) (5.6 ± 1.1 s; figure 6(B), gray; P < 0.05, repeated-measures ANOVA). Closer examination of traversal time of trials using each of the other modes at least once confirmed that they all took longer than using roll maneuvers alone (figure 6(B), colored bars; P < 0.05, repeated-measures ANOVA). The dominance of the fastest roll maneuvers accounted for the animal's high overall traversal performance (79 ± 12% total traversal probability, table 4, figure 8(A), unmodified before; 3.4 ± 0.7 s traversal time, table 4).

3.3.1. Animals locomoting with shells varying in shape

Artificial shells reducing body roundness altered the animal's beam obstacle traversal pathways and decreased traversal probability relative to unmodified body shapes (figures 6 and 7; P < 0.05, repeated-measures ANOVA). Given the results from the unmodified shape, we focused on roll maneuvers and combined all other modes for easier comparison among treatments (see figures A1–A4 for locomotor pathway ethograms showing all locomotor modes). When the oval cone shell was added, the animal less frequently used roll maneuvers alone to traverse the obstacle field (22 ± 5%), likely due to increased volume and surface friction (figure 7(B), red label; table 4). Traversal using roll maneuvers alone was nearly inhibited with the flat oval (3 ± 4%) and rectangle (2 ± 4%) shells due to reduction in body roundness (figures 7(C) and (D), red labels; figure 8(A), red and white together; table 4). This decrease accounted for nearly the entire decrease in traversal probability, because traversal probability using any other mode at least once (figure 8(A), gray) did not change significantly (P > 0.05, repeated-measures ANOVA).

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Artificial shells reducing body roundness also slowed down the animal's traversal (figure 8(B), table 4; P < 0.05, repeated-measures ANOVA). Traversal time for roll maneuvers alone (figure 8(B), white) more than doubled with the oval cone shell (4.4 ± 1.2 s) due to increased volume and surface friction, and tripled with the flat oval and rectangle shells (6.5 s). Nevertheless, with each artificial shell, traversal using roll maneuver alone was always quicker than using any other mode(s) (figure 8(B), gray).

Finally, the animal's obstacle traversal pathways, probability, and time all recovered upon removal of the artificial shells (figures 7(E) and 8, unmodified after, table 4; P > 0.05, repeated-measures ANOVA), confirming that these changes were not due to long-term learning or fatigue.

3.3.2. Animals being pulled with shells varying in shape

Reduction of body roundness also hindered body rolling for deceased animals pulled through the beam obstacles, even when active leg thrusts and postural adjustments were not present. With unmodified shape and the oval cone shell, the animal body frequently displayed rolling as it was pulled through the beams (figure 9(C)). By contrast, with the flat oval and rectangle shells, the animal body almost always pushed forward through while maintaining a horizontal body posture (figure 9(D)).

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Experiments on deceased animals suggested that the constant body vibrations from intermittent leg-ground contact and the ability of the body to pitch up to an appropriate angle appear important in living animals to induce body rolling in combination with rounded body shapes.

Together, these observations demonstrated that, in addition to the performance loss due to increased volume and surface friction of the shells, reduced body roundness decreased the discoid cockroach's capacity to frequently and rapidly traverse beam obstacles by inhibiting the fastest roll maneuvers. This supported our hypothesis that a rounded body shape can enhance traversability through 3D, multi-component obstacles such as grass-like beams.

Our discoveries of terradynamically streamlined shapes in cockroaches inspired a new approach that could enable robots to traverse densely cluttered obstacles rather than navigating around them. Traversing clutter terrain using shape is particularly useful for small robots (Birkmeyer et al 2009, Kovač et al 2010, Hoover et al 2010, Baisch et al 2011, Ma et al 2013, Haldane et al 2015) whose size limits the deployment of onboard sensors, computers, and actuators for real-time obstacle sensing and path planning to cope with such terrain.

To test the shape design inspiration, we challenged our six-legged, open-loop robot, VelociRoACH, to traverse a beam obstacle field similar in configuration to, but larger than, that used in animal experiments (figure 3(B)). We found that with its unmodified, cuboidal body (figure 2(B)), the robot rarely traversed beams, even though its body (6.6 cm) was narrower than beam lateral spacing (10 cm). Instead, as soon as the body contacted a beam (resulting from constant body vibrations due to intermittent ground reaction forces), the robot turned to the left or right, but never rolled, and became stuck between adjacent beams either before entering or within the obstacle field (figure 10(A); movie S2). Only when the robot's trajectory was nearly straight and in the very middle of the two adjacent beams could it run through (figure 12(C)) quickly (1.5 ± 1.8 s) without substantial contact (20% probability, figure 10(C), unmodified; figure A5).

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Upon adding a thin, rounded ellipsoidal shell inspired by our animal experiments, the robot traversed the beam obstacles with higher probability (93%; figure 10(C), ellipsoid slice; figure 11(B)), even though the shell (12 cm) was wider than beam lateral spacing (10 cm). During traversal, the robot first pitched up while pushing against the beams, and then rolled to the side and maneuvered through the gap between the beams (figure 10(B); movie S3) rather than turning after contacting the beams. This was achieved using the same open-loop control without adding any sensory feedback. Further reduction of body roundness using the flat oval and rectangle shells resulted in reduction and eventually inhibition of roll maneuvers (figure 11), reducing traversal probability and increasing time (figure 10(C, D)), again similar to animal observations (figure 7). Together, these findings demonstrated that the mechanical interaction of the thin, rounded body with the beam obstacles alone was sufficient to induce roll maneuvers and facilitate beam obstacle traversal.

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Besides facilitating body rolling via mechanical interaction with the beam obstacles, body roundness also assisted in maintaining the robot's heading. We observed that after the initial contact with the beams, the ellipsoidal and flat oval shells both more often drew the robot towards the middle of two adjacent beams, whereas the flat rectangle shell and the robot's unmodified cuboidal body more often resulted in the body turning away from one beam and towards the other, or even deflection of the robot laterally away from the beams (figures A5–A8).

We noted that the robot's beam obstacle traversal via roll maneuvers was not as effective as that of the animal. The animal traversed beam obstacles with lateral spacing (1 cm) less than half its body width (2.4 cm) and less than its standing height (1.2 cm). By contrast, the robot only traversed beam obstacles with lateral spacing (10 cm) about 80% its shell width (12 cm) and more than twice its standing height (4 cm). This is partly due to the robot's less flattened body aspect ratio (length: width: height = 10:6.6:3) compared to the animal (4.9:2.4:0.7) and its inability to adopt a more sprawled leg posture or slightly flex/twist the body. Furthermore, even with this moderately narrow obstacle gap, many parameters must be well tuned for the robot with the ellipsoidal shell to traverse by roll maneuvers, including the size of the shell relative to beam lateral spacing, the fore-aft position of the shell over the robot body, and the robot's stride frequency. Refinement of robot shell, body, and leg design could further improve its obstacle traversal performance using synergistic operation of body shape, body flexibility, leg morphology, and leg kinematics (Spagna et al 2007).

The robot did not show all the locomotor modes observed in the animal (figure 12). Only four modes resembled those of the animals: roll maneuver, pushing, climbing, and exploration. Moreover, these modes did not closely mirror the animal's corresponding modes (figure 5). (1) The robot often had to push against the beams for a substantial amount of time to pitch its body up substantially before it rolled to the side to maneuver through the beams (movie S3). By contrast, while the animal's body also often pitched up before roll maneuvers, the duration of pitching was briefer (movie S1). (2) During the pushing mode, the robot's legs were always engaging the ground due to the only moderately cluttered beams, while the animal's body and legs were lifted off the ground due to bending of the more densely cluttered beams. (3) The robot's pushing often resulted in a lateral deflection away from the beams. The robot could not turn around and then try to negotiate the beams again like the animal could during exploration. (4) The robot's 'climbing' mode always resulted in a body pitch of nearly 90 degrees against a beam, but the robot was unable to lift off ground and actually climb up the beams, whereas the animals readily lifted themselves and climbed up onto the beams after substantial body pitch. (5) The robot more often flipped upside down (figures A5–A8) due to large pitch and roll vibrations from interaction with the beams, which was only very rarely (<10 out of 869 trials) observed in the animal experiments. The animal could always quickly self-right, whereas the robot could never right itself.

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Together, these differences in animal and robot obstacle traversal modes and pathways suggest that, besides rolling due to the mechanical interaction of a rounded body with the beams, the animal was likely employing sensory motor feedback to further assist traversal. The constant sweeping of antennae of the animal during exploration and beam negotiation phases (movies S1) was likely to detect obstacle openings, guide path selection (Harley et al 2009, Zurek and Gilbert 2014), adjust leg posture and gait, and flex/twist pronotum joint to actively facilitate body rolling to reduce terrain resistance. Recent studies of flight and swimming in analogous 3D environments (Flammang and Lauder 2013, Lin et al 2014) discovered that birds and fish rely on visual and tactile sensory motor feedback to steer and navigate densely cluttered obstacles, rather than following pre-planned trajectories. Further animal experiments of rapid traversal in 3D, multi-component terrain using electromyography (Sponberg and Full 2008) and manipulating the animal's sensors (Harley et al 2009, Mongeau et al 2013), legs, and body flexibility will further elucidate the animal's capabilities and improve robot performance.

To begin to explain our observations of diverse locomotor pathways and the effect of locomotor shape, we developed a preliminary potential energy landscape model of locomotor–ground interaction during beam obstacle traversal (figure 13) inspired by recent success of potential field models for robot manipulation based on contact mechanics (Mason et al 2012). In our minimal model, the locomotor (moving animal or robot) is simplified as a rigid body (ellipsoid) of weight mg with center of mass position (in the Cartesian lab frame) (X, Y, Z) and orientation (Euler angles) (α, β, γ); two adjacent beams are simplified as two massless rigid plates with torsional springs (stiffness = K) at their bases (figure 13(A)). The total potential energy of the system is $E=mgZ+\frac{1}{2}K\left(\Delta {\theta }_{1}^{2}+\Delta {\theta }_{2}^{2}\right),$ where Δθ₁ and Δθ₂ are the angular displacements of the two plates in the x–z plane due to contact with the locomotor. Considering the locomotor as it approaches the beams in the +x direction, Δθ₁ and Δθ₂ are uniquely determined by (X, Y, Z, α, β, γ), thus the total potential energy is a function of both translation and rotation of the body, i.e., E = E(X, Y, Z, α, β, γ). Here we focus on the initial contact phase where the bottom of the body touches the ground. Assuming that the body can rotate freely to access the lowest potential energy at any position in the horizontal plane, i.e., E is minimized over the rotation degrees of freedom (α, β, γ), a potential energy landscape E₀ (X, Y) is obtained (figure 13(B)).

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The preliminary potential energy landscape model began to provide insights into the shape-dependent locomotor–ground interaction during movement in 3D, multi-component terrain. From the potential energy landscape, interaction of the locomotor with two adjacent beams results in two high potential energy barriers around the locations of the beams, with a narrow and much lower potential energy barrier in between. The resulting lowest potential energy orientations of the locomotor are shown at three representative horizontal center-of-mass positions (figure 13(B), insets). When the locomotor is exactly in the middle of the two beams, its lowest potential energy orientation is to roll to its side. When it is closer to one of the two beams, its lowest potential energy orientation is to pitch up. When it is even closer to one of the two beams, its lowest potential energy orientation is to yaw to the left or right. These orientations resemble the observed animal/robot body orientations during the beginning of a roll maneuver (red), climbing (light green), and exploration/deflection (black) (figure 5(A), figure 12 (A, B)). We note that these lowest potential energy orientations are only instantaneous states of the locomotor, and do not fully represent the entire movement pathways (which comprise a series of orientation states). Further, the potential energy landscape is sensitive to locomotor shape. For example, compared to a rounded locomotor such as an ellipsoid (figure 13(B)), an angular locomotor (such as a cuboid with the same length, width, and thickness) results in two higher potential energy barriers with a narrower lower barrier in between (figure 13(D)). In addition, observation of the gradient of the potential energy landscape revealed that the angular cuboid results in repulsive lateral forces away from the lower barrier unless the locomotor is in the exact middle of the two beams (figure 13(D), black curves), whereas the rounded ellipsoid results in attractive lateral forces towards the middle (figure 13(B), black curves).

While beyond the scope of this study, we hypothesize that future developments of such locomotion energy landscapes based on contact mechanics may begin to allow statistical prediction of movement (e.g., probabilistic distribution of locomotor pathways) in 3D, multi-component terrain and understanding terradynamic shapes. The theoretical framework of energy landscapes has been successfully used to understand the mechanisms of fundamental processes and/or make quantitative predictions in both close-to-equilibrium (e.g., protein folding, Wales 2003; DNA origami, Zhou et al 2015) and far-from-equilibrium and/or self-propelled systems (e.g., single molecule pulling experiments, Dudko et al 2006; cellular networks, Wang et al 2008; animal movement ecology, Wilson et al 2012, Shepard et al 2013). We observed that, analogous to statistical distribution of particles in energy states in an equilibrium system, animal and robot movement with rounded body shapes appeared to be more probable via locomotor pathways (e.g., roll maneuvers) that overcome lower energy barriers (the lower energy barrier between the two high barriers; figure 13(B)), as suggested by our preliminary potential energy landscape models. We hypothesize that the constant body vibrations that the animal and robot experienced due to intermittent ground contact during legged locomotion may serve the function of 'thermal fluctuations' to induce body rotation accessing locomotor pathways of lower potential energy barriers.

We note that our potential energy landscape model is only a minimal description of the locomotor–ground interaction physics and does not yet incorporate many possibly key elements, such as stochasticity, dynamics, driving forces, dissipation, and high-level, goal-directed behaviors of robots, and sensory feedback and even cognitive behaviors of animals. Future work is needed to more accurately measure contact physics and validate such potential energy landscape models, and complement them with dynamic simulation, stochastic modeling, models of driving and dissipative forces, and high-level behavioral models.