Brittle stars (Class Ophiuroidea) are an ancient class of echinoderms, with the earliest fossils from the early Ordovician period [1]. Superficially similar in appearance to sea stars (Asteroidea), brittle stars differ dramatically in their locomotion. While sea stars slowly move by using their numerous tube feet while retaining the same overall limb posture [2][3][4][5], brittle stars use large motions of their flexible limbs to climb, grab objects, and push themselves more rapidly along the ocean floor via underwater walking [6,7]. In rare cases they have even been reported to swim [8].

Effective walking locomotion requires acceleration and deceleration of the oscillating limbs and the body, as well as either tolerating frictional losses from dragging the body/appendages or expending the effort to lift these clear of the substrate. Underwater walking imposes further complications, as the buoyancy which supports the body can also lead to lower traction with the substrate and risk of slipping failures [9]. The relative importance of these factors may influence the distribution of mass and, when underwater, dense tissues within the animal. Brittle stars (figure 1(A)) represent a fascinating case to explore, as their pentaradial symmetry enables omni-directional movement [10,11] but constrains their mass and density distribution to a simple proximo-distal gradient along these five limb axes. The dense, calcium carbonate skeleton appears to comprise a greater fraction of the limb volume than the volume of central disk, leading us to hypothesize that the increased density of the limbs will provide a crucial advantage during their unique mode of underwater walking by providing increased traction and reducing slipping.

Brittle star underwater walking differs substantially from other underwater walking species such as crustaceans and salamanders, chiefly in that the brittle star body disk cyclically moves and stops rather maintaining forward momentum. This means brittle stars are subject to substantial accelerations and thus require correspondingly higher ground reaction forces, which is exacerbated by their high density. In contrast to brittle stars, the fastest echinoderms, the fastest terrestrial legged organisms tend to have relatively light limbs and heavy bodies in order to facilitate rapid limb motion [12]. Thus, while lightweight limbs may benefit an underwater walking crustacean or salamander for the same reason as they would benefit a terrestrial walker, heavy limbs may be uniquely advantageous to the peculiar locomotion of brittle stars. Understanding the interplay between limb mass and locomotion has the potential to provide insights into the fundamental principles of underwater walking [13], allowing a better understanding of natural locomotion and enabling better design for benthic robots.

These factors together present an opportunity to leverage recent advancements in soft and bioinspired robotics to create experimental and computational robot models to examine how mass distribution influences speed and optimal gait, a question which cannot readily be addressed via live animal experiments. Previous approaches in experimental robotics [14] and modeling [15] of limbed systems have led to a variety of bioinspired machines [16,17] and unearthed principles of animal behavior [18], ecology [19], and evolution [20]. However, it is only in recent years that we have seen these experimental and computational tools extended to the domain of soft-bodied robotic systems [21][22][23]. For organisms with highly flexible appendages such as the brittle star, soft robotics provides a unique framework for creating model systems that analogize the morphology and the intrinsic mechanics of their natural counterparts without the need to fully reproduce the complexity of their high degree of freedom limbs [24].

In this study, we use soft robotics and emerging methods in differentiable computational modeling for contact-rich scenarios to examine the influence of density distribution in the gait and speed of organisms that perform rowing-like behaviors. We postulate that the relatively high density of the limbs is advantageous for brittle star-style underwater walking due to a beneficial reduction in the actuation effort expended to remain in contact with the ground relative to the effort necessary to move the mass of the body. Thus, given some traction mechanism, we predict less actuator energy is required to keep the pivot limbs in contact with the ground if the mass in the limbs is greater and, additionally, less energy is needed to move the rest of the body (because the mass of the body is less). Note that throughout the article, we consider an idealized case in which friction is the only force resisting slipping. Although this is a simplifying assumption that abstracts from real methods of traction, such as ambulacral feet, the abstraction allows us to study the physics related to the mass distribution in a simplified setting. To investigate this, we perform the following series of complementary studies. First, measurements are collected from euthanized brittle stars to determine mass and density of different body parts. Next, experiments are performed on soft, untethered biomimetic robots of varying mass/density configurations in order to systematically test the effects of mass distribution on locomotion (figures 1(C) and (D) show the robot). Finally, two theoretical models are used to investigate the nature of the experimental observations. The first is a simple static model, which illustrates that mass distribution in brittle star morphology significantly effects traction mechanics. The second is a more complex dynamic simulation. Leveraging the deep connections between optimal control theory, mechanics, and morphology [25][26][27][28], we use iterative linear quadratic regulator (iLQR) [29] to validate our hypothesis and to investigate root causes.

Most brittle stars have five limbs (also often called arms), and exhibit pentaradial symmetry (figure 1(A)). As shown in the figure, these limbs radiate from a central body disk, which contains the mouth along with internal organs. Brittle star sizes can vary widely, with limb lengths ranging from 0.3 to 60 cm [30]. Their mass similarly varies, but specimens in this study were approximately 1-10 g (with limb lengths roughly 5-10 cm).

In order to move, brittle stars choose a direction of travel and adopt a bilaterally symmetric, coordinated gait called rowing [5,31]. A very similar gait, called reverse rowing, is also sometimes used [5,7], but in this work we focus on rowing due to its greater prevalence. Briefly, the animal chooses the limb pointing in the desired direction of travel to be the 'leading limb' , then the two adjacent limbs perform a rowing motion, swinging forward, pushing down on the substrate, and then rowing back to push the animal forward [32,33]. Note that the leading limb is a temporary designation and the specific limb used as leading limb frequently changes. As the brittle star executes this motion, the (momentarily) rear limbs may also 'kick' against the substrate, potentially adding propulsion (although this has yet to be proven with ground reaction force data) [33]. After this motion, the body disk comes to rest on the substrate while the limbs are re-positioned [33]. Figure 1(B) shows a diagram of the gait. See Astley [33] for a full analysis of the kinematics. This locomotion is defined as underwater walking, but differs from other taxa due to the low importance of hydrodynamic drag (due to the flattened body disk and relatively low speed) and relatively massive limbs [34,35]. These differences make the empirical and modeling approaches described in the cited works difficult to apply to this organism.

To discern whether brittle stars have a noticeable pattern of density and mass distribution throughout ). a % Weight in Arms denotes the underwater weight their bodies, five individuals of five different species were acquired from the aquarium trade and euthanized. Whole animals were weighed, both in air and in water. These can be used to calculate the bulk density of each animal. After getting these values for the whole animal, each animal was dissected by cutting off the limbs at the base of the limb where the disk and limb meet. Then, each of the five limbs and the disk were weighed, again both in air and in water, to obtain densities of these separate parts. The data from these brittle stars are shown in table 1. The most striking trend is that the limbs are substantially more dense than the discs and tend to contain about 70%-80% of the total mass and a comparable value of the underwater weight. The final row in the table designates the mean µ and standard deviation σ. Performing a standard t-test on the density data, we reject the null hypothesis that the arms and disks belong to the same normal distribution with p = 0.00023.

It is not necessarily intuitive why the distribution of mass would change performance (e.g. distance moved per stride, cost of transport, etc) of rowing organisms or analogous robots during locomotion. Several reasonable mechanisms can be postulated. One possibility is that distally distributed mass provides a larger support polygon throughout locomotion [36]. We will show that, while this is true, it is not the primary limiting factor for the models we examine. Another consideration is that a heavier central disk will result in more frequent contact between the disk and the substrate, resulting in additional friction that opposes locomotion. However, if the mass is distributed out to the limbs, then the inactive limbs could also be a potential source of added friction, eliminating some of the possible benefits. Conversely, it is possible that the favorable traction conditions induced by more distal mass distributions plays an additional role in aiding the locomotion. Specifically, the limbs must engage with the substrate during the power stroke in order to efficiently push the rest of the body forward; any slipping (or alternatively, effort or energy expended to maintain contact) results in less efficient forward motion. Therefore, stick-slip behavior is critical. Here, we discuss two modeling approaches that examine this stick-slip mechanism of the rowing locomotion problem.

We first use a simplified analytic model to examine stick-slip in the limbs and its relationship to weight distribution. We assume that the brittle star has five limbs. For cases with three or four limbs, we refer the reader to the first section of the SI, where we generalize the following model to those cases. For cases with less than three limbs, the brittle star can no longer statically support itself and thus the locomotion fundamentally changes (see Kano et al [37], which includes excellent videos of this). We do not study these cases in this work, however, we note that in such cases mass distribution should become even more important, because the arms now have to drag the body along the ground. The brittle star is represented by a quasistatic model in which the entire mass of the body is concentrated in a radial center of mass (COM) that is determined by the mass distribution. The geometry of the model is shown in figure 2(A). The red circles in the figure represent the location of the radial COM. We anchor the x = 0 point of the coordinate system to the location of the central disk. The limb exerts a tangential force, F t , on the substrate which must be resisted by a traction force if slipping is to be avoided. Here, we represent the traction as friction, which is the simplest case, but the qualitative findings of our model apply to any traction mechanism that depends on the preload magnitude, in this case the Normal Force, N. Thus

where µ is the coefficient of friction and N 1 is the normal force on the rowing limb tip. We make the simplifying assumption here that the limb makes contact at a single point at the end of a limb. While real brittle stars make contact at some intermediate point (or segment) on the limb, we can think of this model as essentially ignoring the component more distal than that contact point. Geometric quantities include the limb length ℓ, the COM location with respect to the center of the disk ℓ c , and the coefficient c, which represents the distance along the limb at which all mass in the limb is concentrated. Thus c = 0 corresponds to all mass being concentrated in the central disk while c = 1 corresponds to all mass located in the tips of the limbs. This is an extreme case, but the model is useful for examining the fundamental mechanics.

Because we assume that locomotion is quasistatic, the fundamental feature governing the stability of locomotion is the support polygon [36]. In short, a robot or animal can statically support itself as long as the COM remains in the convex hull of the contact points. This region is often referred to as the support polygon. This is the geometric limit of a quasistatic legged gait. In our case, we examine forward locomotion and therefore the COM will always leave the support polygon in the forward, or +x direction. In other words, when the COM moves past the contact point of the lateral limbs (at distance ℓ 1 from the central disk in figure 2(B)). Therefore, the COM can be said to be in the support polygon when

where ℓ c is the location of the COM, ℓ 1 and ℓ 2 are the planar distance between the central disk and the point of contact of the lateral and posterior limbs respectively. Here, ℓ 1 = ℓ cos(θ) and ℓ 2 = ℓ cos(π/5) (the neutral position for the rear limbs) where ℓ is the length of a limb and θ is the angle of the rowing limb of length ℓ 1 with the axis of locomotion (labeled +x in the figure). So, for a small θ, the rowing limb is parallel with the x axis and for θ = 90 • it is perpendicular to the x axis.

Next, looking at the free body diagram in figure 2(B), we resolve the statics by taking the sum of moments about the point x = -ℓ 2 as follows:

to find the normal force, N 1 , as a function of c and θ,

where m is the mass of a single limb, or one fifth of the total mass. Determining this normal force is crucial, as a greater normal force implies a greater ratio µN 1 (c, θ)/F t , which, from equation ( 1), can be thought of as a margin of safety against slip. A greater ratio µN 1 (c, θ)/F t in turn suggests that a given body plan, characterized by c, is more robust to slipping and less control effort needs to be expended to maintain traction. To quantify this, we will normalize N 1 (c, θ) with respect to the case N 1 (c = 0, θ). This dimensionless quantity will allow us to characterize how mass distribution c effects normal forces throughout the duration of the gait cycle characterized by varying θ. First, we examine the case where c = 0 (all mass in the central disk). We find We can now use this value to construct our dimensionless ratio:

If this ratio is greater than unity,

and thus from equation ( 1) is less likely to slip.

We plot (2) in figure 2(C). This establishes the point at which a stable quasistatic stance can no longer be maintained for a given mass distribution based on the support polygon. In figure 2(D), we plot the ratio in (6) for several mass distributions c and we include the line from ( 2) that provides the limiting stroke of a normal gait cycle. For the majority the stroke, more distributed mass configurations (i.e. larger c) have larger normal forces, which implies that configurations with more distal radial centers of mass are less likely to slip in those regions. This relationship flips beyond the neutral configuration of the lateral arms (θ > π/5). However, the majority of the power stroke, particularly during the earlier parts of the stroke while normal forces are at their maximum, occurs under conditions where distal mass distributions are more favorable towards maintaining traction because N 1 is relatively larger. As we will see in the next section, this additional margin for error induced by more favorable static conditions allows more efficient movement for the entire gait cycle.

To validate that this phenomenon holds for dynamic situations, we used a recently developed physics engine, Dojo [38], to simulate robots/brittle stars with various distributions of mass. The chief advantages of Dojo for this work are its combination of physically real contact model and differentiability, allowing rapid simulation and optimization of contact rich problems like the one at hand. We first compare velocity across different mass distributions for given sequences of inputs (in other words, a fixed input budget). Then, to validate our findings in more realistic scenarios, we compare velocity across mass distributions of optimal trajectories (this time using standard methods to implicitly optimize the cost of transport). The algorithm used for optimal gait design is iLQR, which exploits the Bellman optimality of the linear approximation of the trajectory optimization problem to rapidly find solutions. We briefly elaborate on the scientific value of each of these sets of experiments. First, the controlled setting with identical inputs u t at each timestep allows us to directly compare the performance of cases with different mass distributions, letting us search for clues about the mechanisms underlying the differences in performance. Conversely, as we will see, trajectory optimization results do not allow as much clarity in our search for underlying mechanisms. This is because the optimizer converges to different gaits, compensating for the shortcomings of an inferior body plan by adapting in the controller. However, it is still valuable to our analysis, because it allows us to answer the question: 'What if different sets of inputs are better for for different body plans, and does that result in different overall conclusions in terms of the effectiveness of different body plans?' So the main value in the optimization is the fact that we can examine our results to find if certain mass distributions remain more effective, even when we optimize inputs for each distribution.

Briefly, the robot model is specified in simulation as follows. First, a branching structure is defined in which there is a central disk and five evenlyspaced limbs. Each of these limbs is composed of two rigid links connected by a spherical joint, and an identical spherical joint connects the proximal link of each limb to the central disc. The joints have both stiffness and damping to crudely simulate the flexibility of the elastomeric robot limbs (as in [39]). Hydrodynamic drag is neglected due to the low speed of the locomotion of the animals, simulations, and robots (1-3 cm s -1 in [33], < 2 cm s -1 , and < 0.5 cm s -1 , respectively). Observed mean drag forces during underwater walking of axolotls and prawns were < 10 mN despite being much larger animals moving considerably faster (7.7 cm s -1 and 4.4 cm s -1 , respectively) [9], far smaller than the forces in the simulation (figure 3(C)). The robot is represented in minimal coordinates, with the state vector consisting of the position r b , orientation ϕ b , linear velocity v b , and angular velocity ω b of the central disk in Cartesian space and the set of joint angles and velocities in minimal coordinates [θ x i , θ y i , θx i , θy i ] for joint i (note that the joints do not twist, they only bend). The full state is then a fifty-two element vector:

One note on the representation; while this is the state that the user interfaces with in this work, Dojo uses maximal coordinates to represent rigid bodies and quaternions to represent rotations [38]. Finally, the inputs in this model is the vector of generalized torques, u, acting on the joints.

To compare different mass distributions, we specify four cases in which we simply vary the percentage of mass in the limbs (with mass evenly distributed between distal and proximal segments) versus in the central disk and perform two analyses on this set. We also compare these cases for short (130 cm) and long (260 cm) limbs. First, a simple, hand tuned gait is created to perform open loop rollouts in simulation. For each case, we use the distance traveled as a measure of relative performance. From figures 3(A) and (B), it is clear that configurations with distal mass configurations result in better overall performance, although this relationship is not monotonic. Snapshots for a representative optimal gait are shown in the figure inset.

To determine the mechanism that induces this behavior, we can look at the contact forces. For the same rollouts, figure 3(C) shows contact forces on the tips of the active limbs. These frictional forces are the propulsive forces that are being generated by the rowing limbs; higher friction here means larger traction forces can be sustained. Also, if slipping occurs, it will show up here. If we look closely at the beginnings and ends of the contact periods for the central case, it is apparent that slipping is occurring as the limb briefly makes and breaks contact with the substrate. The frictional forces sustained by the central case are also considerably lower here. Taking the integral we find total impulses of 1.5380N • s for the distal case and 0.8867N • s for the central case, a difference of almost a factor of 2. We can also look at the positions of the arm tips in figure S1(A), which shows that the distal case holds its position substantially better, while the central case slides more through most of the power stroke.

We examine two additional possible mechanisms that could explain differing velocities. Figure S1(B) shows contact forces on the body and inactive limbs (i.e. excluding the rowing limbs) from two of the rollouts (98% of the in the limbs, referred to from here as distal, and 14% of the mass in the limbs, referred to from here as central). The frictional force from this plot gives a notion of the contact friction resisting locomotion during the rollout. From the plot, we can see that the central case (labeled 14%) has slightly higher friction. Indeed, taking the area under the dashed curves, we find a total impulse of 3.4608N • s for the distal case and 4.0161N • s for the central case, confirming our intuition that there would be slightly more dragging in the central case.

Finally, we should make a note on the polygon of support. As a reminder, based on the elementary statics from the previous section, the distal case should have a more forgiving support polygon. However, we observe that the differences in velocity occur primarily during the time period of the gait where the COM is in the polygon of support. This is shown in figure S2, where we observe that the body leaves the support polygon at approximately identical points for both extreme cases. This leads us to confirm that the critical difference in outcome is influenced more heavily by the differences in traction conditions during the supported phase (i.e. while the COM is in the support polygon).

We next perform experiments using iLQR to compare performance between the different distributions while executing a more realistic gait. The reference gait and reference input trajectory (also designed with iLQR) are fed as input to the iLQR problem for each configuration. The optimizer is tasked with replicating this reference state trajectory while minimizing the inputs. The cost function at each time therefore takes the following form:

where x t and u t are the state and input vectors defined previously, x ref t is the state of the reference trajectory, Q t is the (diagonal) state cost matrix, and R t is the (diagonal) input cost matrix for time step t. Similarly, the cost for the final step is

where Q f is the state cost matrix for the final time step. For these specific trajectory optimizations, (10) and

where I n is the n × n identity matrix. The forward progress along the trajectory is shown in figure 3(D). Performance for the central mass case is much worse than for the other cases, as expected based on previous hypotheses. To determine whether stick-slip behavior is involved, contact forces can be used as before. Figure 3(E) shows the contact forces on the body and inactive limbs and figure 3(F) shows the propulsive forces from the active limbs. While figure 3(F) is less revealing visually than figure 3(C), we can still look closely and note that the largest difference in acceleration (figure 3(E)) occurs around the same time where the difference between the traction forces (figure 3(F)) is the greatestduring the peak around 0.4 s. but we can again use integrals to find contact impulses. For the friction on the body resisting motion, the impulse is 0.6793N • s for the distal case and 0.7846N • s for the central case. For the friction supporting the active rowing limbs, the impulse is 0.6308N • s for the distal case and 0.5281N • s in the central case. While the differences are more subtle than before because the optimizer compensates well, these impulses suggest that the central case cannot use as much force to push itself forward. Note that in these simulations, slipping does not actually occur because the trajectory optimizer avoids it.

Finally, for a third set of simulations, the same reference input trajectory is used as a starting point for the forward rollout of iLQR. However, this time the cost for not following the reference state trajectory is essentially non-existent and instead the cost function is designed to maximize forward movement while minimizing actuator magnitudes. Therefore, while equation ( 8) still holds, the reference state trajectory is redefined to have a much more sparse structure:

, 0, z 0 , 0, 0, 0, 0.5, 0, 0, 0, . . .

where z 0 is the height of the robot. Therefore, over the course of the trajectory, the optimizer is free to explore a wider range of behavior. Similarly, at the final step, the reference trajectory is

0, 0, z 0 , 0, 0, 0, 0.5, 0, 0, 0, 0, 0, ×θ 1,0 , θ1,0 , θ 2,0 , θ2,0 , . . .

where θ i,0 is the (2×1) vector of angles for joint i at t = 0. This simply induces the optimization to find trajectories that end in about the same configuration as they started, which is desirable so we avoid motions that are not realistic for a periodic gait. This method results in the optimizer finding the (local) best trajectory for a given mass configuration and environmental friction. Theoretically, if configuration does not matter, these should perform roughly the same. However, the results of these experiments show large qualitative and quantitative differences in optimal trajectories. Snapshots from two such optimized trajectories are shown in figures 4(B) and (C) and the corresponding videos are Movies 2 and 3. Figure 4(B) shows a trajectory optimized for a brittle star-like configuration (75% of the mass is in the limbs) in a high friction environment. The resulting trajectory is qualitatively similar to the brittle star, exhibiting rowing behavior as in [33], along with a commonly observed supplemental kick with a 'rear' limb at the apex of the row. This is the fastest trajectory found in our experiments. Conversely, figure 4(C) shows a trajectory optimized for the case where only 15% of the mass is in the limbs (in a low friction environment). It begins with rowing-like behavior, but it notably begins repeatedly bouncing forward on its limbs about 0.67 s into the gait. This behavior, usually referred to as punting when it is aquatic, has been well studied in the robotics and animal locomotion literature [40][41][42]. Calisti et. al make use of an extension of the SLIP model, called Underwater-SLIP, to describe it rigorously [35]. One assumption of the model is that the mass in the legs is small compared to the body, which is a reasonable assumption for figure 4(C) but not figure 4(B).

The purpose of testing under variable friction coefficients is to both study how differing conditions result in different optimal gaits, but also to verify that our conclusions do not break down in drastically different environmental conditions. Quantitative results from these experiments in figure S3 show that configurations with more mass in the limbs tend to perform better than configurations with more mass in the disc, regardless of the environmental coefficient of friction chosen for the simulation, in line with our previous hypotheses and data. An additional notable feature of this data is that at lower friction, the 98% mass in limbs case performs the best, but at higher friction, the case with 75% performs the best. One possible interpretation of this result is that in a situation with better environmental traction, there may be diminishing returns to improving traction with heavier limbs, while the cost of moving the limbs through the swing phase depends only upon their mass and geometry. The snapshots in figure 4(B) correspond to the 75%, µ = 0.5 case of figure S3 (dashed orange line), while those in figure 4(C) correspond to the 14%, µ = 0.1 case of figure S3 (solid purple line).

Having verified that brittle stars tend to have much more mass in the limbs and having shown using multiple computational approaches that this provides an advantage, we next seek to experimentally characterize the effect of mass distribution using a physical robot model. To test the effect of varying mass on In all cases forward, here called the x direction, is from left to right. (A) A simple, hand-tuned brittle star-like rowing gait performed by a simulated brittle star with 75% of its mass in the limbs. This is quite similar to the gait used later for on-robot experiments. (B) A trajectory optimized for a simulated brittle star with 75% of its mass in the limbs. This gait was found by loosely constraining the iLQR algorithm to 'go forward' , 'do not jump' , and 'end in a similar configuration to the start' all while minimizing the actuation force. The resulting gait has qualitative similarities to brittle star gaits described in the literature [33]. (C) A trajectory optimized for a simulated brittle star with 14% of its mass in the limbs. In this case, it is notable that the solution exhibits punting behavior from about 0.67 s to 1.63 s.

brittle star-like underwater walking, we created a biomimetic brittle star robot based on material architectures and limb designs explored in previous work [43][44][45]. While the design of the robot necessarily makes simplifications and trade-offs to accommodate engineering constraints, it is designed to be biomimetic in several key senses. A five limb design is chosen to reflect the body plan of most brittle stars, with limbs radiating out from a central disk at 72 • intervals. Since flexibility is a core feature of brittle star morphology, this robot is a continuum soft robot, meaning that many of its critical structures, most importantly the limbs, are made of flexible materials. We make this design choice because brittle star limbs are highly articulated and exhibit a high degree of flexibility. The robot limbs contain 'artificial muscle' actuators composed of coils of nickel-titanium shape memory alloy (SMA). The coils are arranged into the four quadrants of the cross section and run through the length of the limbs. While the brittle star has muscles arranged in similar fashion, it has many sets, each corresponding to a joint between adjacent vertebral ossicles in the limb. We chose to idealize this morphology as a single set of four muscles because of the difficulty of scaling power management, communication, and fabrication for many more actuators within an untethered, wireless robot testbed.

To test the robot's performance with different densities and distributions of densities, we needed a simple but unobtrusive way to add and subtract dense and light material from the robot. To accomplish this, we use steel attachments and foam. To add density to the limbs, steel rings are placed around it at specified locations. These locations are at the base of the limb, referred to as a proximal addition, at the middle of the limb (medial), and at the tip of the limb (distal). To add density to the central disc, steel plates are placed into a fixture on top of the disc. Finally, to remove density from the disc, foam is affixed.

To systematically test the effect of varying density distribution, we choose a simple test wherein the robot performs a specified gait for several cycles over a sandpaper-like surface at the bottom of a fish tank. Position data is captured and used to calculate average velocity and a cost of transport estimate. The gait is designed to mimic the rowing gait of the brittle star, although due to the robot's relatively small number of degrees of freedom and to its lack of proprioception, the resulting gait is comparatively simple. For a graphical visualization of the rowing gait, refer to the diagram in figure 1(B). Since the executed gait is identical in each case, density distributions with higher average velocity are considered superior.

Snapshots of representative experiments are shown in figure 5(A). Average velocity for all trials is plotted against radial COM and net weight in figure 5(B). Figures 5(C) and (D) break out the same data against radial COM and net weight respectively. Qualitatively, it is clear that radial COM is positively correlated with velocity for low to medium radial COM (i.e. more central mass distribution) while the correlation stops for higher values (i.e. more distal mass distribution). Net weight is, predictably, negatively correlated with velocity. For statistical analysis, we take as dependent variables the net weight as a discrete variable with 5 levels and the radial COM as a nested continuous variable. The average velocity per gait cycle is taken as the independent variable. We then perform analysis of variance (ANOVA) and find that both net weight and radial COM have statistically significant effects in our sample (p < 0.01). In terms of effect sizes, we find that the net weight explains 43% of the variance of our data while, after fixing the net weight, the radial COM explains 6%.

However, as can be observed in figure 5(C), the correlation appears much stronger at a lower radial COM. Indeed, restricting the analysis to the lowest third of the group in terms of radial COM, the effect sizes become 22% for net weight and 69% for radial COM (p < 0.01).

Based on the computational and experimental studies with the robotic model, it is clear that the mass distribution observed in brittle stars (in which 70%-80% of the animal's mass is in the limbs) plays an important role in locomotion. The results compiled for this work collectively support our hypothesis that the primary mechanism through which mass distribution influences brittle star style walking is through reducing the traction forces necessary to fix the propulsive limbs to the substrate. Future work could further test our results in a comparative context, examining mass distributions across Ophiuroids of different habitat and activity levels and inferring such distributions from fossil taxa.

However, the experimental and modeling results raise an interesting question: is a more distal mass distribution always better? The results presented suggest that more distal is likely no better than the distribution seen in the brittle star (70%-80% of mass in the limbs). In fact, on the robot, locomotion speed appears to increase regardless of where the mass is placed on the limbs. Specifically, in figure 5 we observe that the slowest cases overwhelmingly occur when mass is added to the central disk of the robot and when these cases are removed there is only a weak correlation between mass distribution and velocity. Likewise, trajectory optimizations performed in simulation suggest that while it is still better to have most of the mass in the limbs, it is not necessarily optimal to have it as distally concentrated as possible. This is particularly true under low slip contact conditions with high friction, as shown in figure S3.

Although we did not calculate the radial centers of mass of the brittle stars, we can clearly see from data shown in the SI (and from looking at the volumetric taper of most brittle stars) that the majority of limb mass is concentrated in the proximal segment of the limbs. This may be due to biological factors unrelated to the mechanics of locomotion (such as developmental constraints or other limb functions), but based on data collected in this study, this distribution is mechanically beneficial for their locomotion. Further studies incorporating brittle stars of diverse arm morphologies and locomotor behavior in an explicitly evolutionary context are needed to determine whether arm density has been subject to selective pressure for locomotor performance. This work also has implications for the study of other forms of walking. Prior work has begun to investigate extinct echinoderm locomotion through a robotcentric framework [46]. Connecting our results with the rich paleontological record of brittle stars and other echinoderms is an exciting avenue for future work to investigate the influences of basic locomotion considerations through evolutionary history.

From the perspective of robotics, our work has important takeaways as it relates to the design of legged benthic and, more generally, crawling robots. For similar forms (excluding those with very different dynamics, such as crabs [47]), the results of this study can guide robot design from the ground up. Our robot serves as an illustrative example, as it was designed before undertaking this study. An appealing redesign of the robot in light of this work would be to do away with the central battery and add smaller batteries to the proximal part of each limb. More generally, we can sum up the most important design takeaway in a single, simple maxim: distribute heavier materials onto the limbs while keeping lighter ones on the body. The full class of morphologies to which this advice can be applied has not been studied, but based on our computational analysis it will likely apply to any crawling robot that uses friction or similar adhesion mechanisms to perform 'rowing' type locomotion (so, for example, we can rule out hexapod robots [48] but recent hatchling sea turtle robots likely exhibit similar dynamics [49,50]).

While our models provide substantial evidence towards our hypotheses, they could be improved in two key ways. First, we made the assumption that brittle star locomotion is quasi-static and takes place in the boundary region near the fluid substrate interface and therefore that the contribution of hydrodynamics are marginal. Removing this assumption would allow investigation into the interactions between hydrodynamics and contact mechanics. Second, the literature has several other models of underwater walking of other organisms such as lobsters [34], octopuses [35], axolotls [9], and searobins [51]. It would be worthwhile to compare and unify these models to be able to make broader generalizations about underwater walking. This is especially relevant in light of the results from our trajectory optimizations with relatively light limbs, which converged to a punting gait similar to the one discussed by Calisiti et al [52]. Based on this variety of results on underwater walking, there are likely multiple active and passive strategies for mitigating slip and generating propulsion [9].

In conclusion, this work presents animal data showing that the limbs of brittle stars are heavier and denser than the central disk. Performing experiments on a brittle star-like soft robot, the relationship between variations in the distribution of mass and robot performance are characterized, demonstrating that more mass in the limbs is much more effective than more mass in the central disc. Finally, two theoretical models are used to investigate the root cause of this relationship. The models suggest that stickslip behavior is paramount and that more mass in the limbs helps to avoid slipping. This work brings a physics-based perspective to the study of brittle star locomotion and behavior and hopefully will provide a new tool to paleontologists and evolutionary biologists to understand these fascinating creatures. robotic model in the Dojo simulator. The robot is given a central disk and five limbs, each containing two rigid segments. All joints are subject to small stiffness and damping. The mass is set to a constant value. To vary the mass distribution, we can simply change the mass parameter of the limb segments and the central disc. To perform initial simulations, a simple trajectory of inputs is determined by trial and error to arrive at a heuristic analog to the brittle star rowing gait. After collecting data to compare performance for different mass distributions, this trajectory is fed as an initial guess to an iLQR algorithm. In order to encourage the algorithm to pursue good gaits, a relatively sparse cost function is specified: the only terms that are included are the body's forward position and velocity (to encourage forward movement), the body's vertical height (to prevent jumping), the final state of the limbs (we want a cyclic gait so we force them to finish in the same state as the starting state), and the inputs (to minimize cost of transport).

The resultant gait is used as a reference trajectory for a new trajectory optimization. This time, the cost function penalizes straying from the reference trajectory. For each mass configuration, the iLQR algorithm attempts to follow the trajectory. More effective configurations follow the trajectory more accurately for lower actuation cost. Finally, this reference trajectory is used as a starting guess to perform another set of trajectory optimizations. Here, the cost function is set up with the same sparse configuration as before. iLQR is then performed for each mass distribution. Code is available at https://github.com/ zpatty/weight_distribution.

To get mass and density information from brittle stars, five specimens from a commercial aquarium dealer were euthanized in a solution of 2 g l -1 MS222. Following euthanization, the animals were dabbed dry with paper towels and weighed intact on a tared balance to obtain the mass. To obtain the volume, a beaker of distilled water was placed onto another balance, which was then tared. A small kevlar string was tied loosely around the animal and then lowered into the water, ensuring that the animal was fully submersed and not touching the wall or bottom of the beaker. The value on the balance was the mass of water displaced by the object, which corresponds to volume assuming a water density of 1 gram per ml. To find the density, one can simply divide the mass by the volume. Next, the animal was dissected by cutting the limbs off where they meet the central disc. Then, the above weighing procedure was repeated for each limb and for the disc.

The brittle star inspired robot has a central disk and five limbs. Each limb is a 16 mm diameter cylinder with four SMA (springs (0.015in Dynalloy 90 • C Flexinol® with a 0.1in helical diameter) arranged in quadrants around the cross section. These SMA springs serve as 'artifical muscle' in lieu of motors or other actuation technology. The SMA is actuated by inducing a voltage drop across it, causing phase transition via Joule heating. The limbs are approximately 14 cm. Near the central disc, the limbs widen interface with the disc. Each limb contains a small PCB with MOSFETs that control the SMAs and a connector to attach to the central circuit board. The central disk contains the microcontroller, an nRF52832 in a Laird BL652 module. This module communicates wirelessly via bluetooth with an offboard receiver, which transmits control instructions from a computer. The robot is controlled by a custom Python script that executes open loop gaits. For power, the robot uses a 7.4 V drone battery.

The limbs are fabricated by injection molding in a 3D printed mold (Stratasys VeroWhite). The SMAs are attached to the limb MOSFET board and placed into steel tubes, which are then sealed with a small amount of adhesive to prevent silicone from entering the tubes. These tubes are placed in 3D printed inserts, which hold them in the correct places in the mold. Then, Smooth-On DragonSkin 10 silicone elastomer is injected, overmolding the PCB and 3D printed inserts. The part is then demolded and the SMAs are crimped, together, to central wire which travels through the center of the limb from the PCB. Finally, the part is placed in a new mold and the tip is overmolded onto the end. The central disk is fabricated by simply placing the central PCB in a mold and injecting the same silicone. The final weight of the robot, battery included, is 320 g. Since the battery is the heaviest part, it is encased in Smooth-On Soma Foama 15 to cancel out its weight in water.

To vary the effective weight distribution of the robot, steel weights are added to the limbs and to the central disc. To add mass to limbs, steel rings (22.225 mm OD, 16.13 mm ID, 2 mm thick) are placed around each limb at one of three locations along the limb (proximal, medial, distal). Each ring is about 3 g. Steel discs (30 mm diameter and 2.5 mm thick) are used to add weight to the center of the robot. Each is about 15 g. Finally, reduce the density of the center of the robot, foam discs of known buoyancy (0.08 N) can be added under the battery.

Thirty-four experiments are performed while varying the total mass and COM using the above methods. In each experiment, the robot is placed in a water tank onto a piece of Rock-on-a-Roll, a sandpaper-like artificial rock surface. Then the robot begins to perform an open loop rowing gait while an overhead camera films. After filming, computer vision is used to extract the location of the center of the robot to extract locomotion information.

For statistical analysis, data is broken up into discrete gait cycles corresponding to the set gait period. We performed a 2-way ANOVA where independent variables are radial COM and net weight and the dependent variable is the per-cycle average velocity. Radial COM is considered a continuous variable and net weight is considered a discrete variable with eight levels. We implement the ANOVA in MATLAB using the anovan() function. To measure effect size, we calculate η 2 using the sum of squares of residuals.

Zach J Patterson  https://orcid.org/0000-0003- 4371-7442 Henry C Astley  https://orcid.org/0000-0003- 0136-1433 Carmel Majidi  https://orcid.org/0000-0002-6469- 9645