Soft skeletons transmit force with variable gearing

A diverse array of biological structures move by transmitting mechanical work through a soft body. This is achieved with hydrostatic skeletons, which are liquid-filled, pressurized structures that are actuated by layers of muscle. Examples include the human tongue, the body of worms and cnidarians, the mammalian penis (Chapman, 1958; Kier, 2012), and the body of a muscle, which itself generates intramuscular pressures that affect force transmission (Azizi et al., 2008; Kier, 2020; Sleboda and Roberts, 2020). Thus, hydrostatic skeletons are ubiquitous among animals and our understanding of their properties informs the field of biologically inspired soft robotics (Kim et al., 2013; Rus and Tolley, 2015; Laschi et al., 2016; Hawkes et al., 2021). The study of hydrostatic skeletons has demonstrated how their geometry mediates the transmission of muscular displacement (Cowey, 1952; Clark and Cowey, 1958; Kier and Smith, 1985; Kier and Van Leeuwen, 1997; Van Leeuwen and Kier, 1997). However, it remains unclear how the transmission of force, and hence mechanical work, depends on the geometry and fiber winding of hydrostatic skeletons.

The gearing of a skeleton determines the extent to which muscular work is applied to displacement or force. Gearing is quantified by mechanical advantage (MA), the ratio of output to input force, and displacement advantage (DA), the ratio of output to input displacement. For a rigid skeleton, the MA about a joint may be shown from a balance of torques to equal the ratio of in-lever to out-lever lengths (Fig. 1F), which equals the inverse of DA (Smith and Savage, 1956). Neither metric changes with joint rotation and thus MA and DA are fixed properties of a skeleton's geometry. As a consequence, measurements of MA and DA have routinely offered a basis for interpreting the functional morphology of vertebrate skeletons (Alexander, 1983; Biewener, 1989; Dunn, 2018; Westneat, 2003).

Rigid skeletons composed of multiple joints demonstrate how skeletal systems may change gearing as they deform. For example, the stomatopod raptorial appendage possesses a 4-bar linkage system that initially has a high MA that contributes to its rapid acceleration. As the strike advances, a reduction in MA enables input work to be imparted over a large displacement (McHenry et al., 2012). Flea legs similarly exhibit a high MA early in a jump as the body accelerates (Bennet-Clark and Lucey, 1967). In contrast, the initial low MA of a frog's leg allows for storage of elastic energy prior to the principal body acceleration, but then increases over the jump for high displacement and work output (Astley and Roberts, 2014). Therefore, a change in gearing over the motion of an appendage has the potential to enhance elastic energy storage, force output and/or speed.

Geometric models of soft skeletons suggest that they also have variable gearing. Squid tentacles are a classic example of variable gearing. These tentacles capture prey by rapidly accelerating forward through contraction of circular and transverse muscles that pressurize the structure's interior and reduce its radius. For a constant-volume cylinder, length varies with the inverse square of radius. Consequently, the DA of a tentacle increases rapidly as it reaches full extension (Kier, 1982; Kier and Smith, 1985; Kier and Van Leeuwen, 1997). The extent to which variable gearing is a general feature of soft skeletons and whether MA is inversely related to DA, as previously proposed, remains to be determined (Kurth and Kier, 2014, 2015).

A complicating factor in the geometry and mechanics of soft skeletons is the common presence of a superficial layer of connective tissue. The connective tissue layer generally features a cross-helical arrangement of stiff collagen fibers that contribute to force transmission while providing reinforcement against aneurysms under pressure (Clark and Cowey, 1958; Wainwright, 1988; Shadwick, 2008). The helical arrangement factors into how these structures deform because a helix encloses a cylinder with a radius, length and volume that varies with its pitch (Fig. 1C,D; Cowey, 1952; Clark and Cowey, 1958). The pitch, expressed by the helix angle θ, conforms to accommodate its interior volume.

The tube feet of echinoderms, used by sea stars for locomotion, offer an example of a variable-volume hydrostatic skeleton with cross-helical fibers. A tube foot (i.e. podium; Smith, 1947; McCurley and Kier, 1995; Leddy and Johnson, 2000; Ellers et al., 2021) consists of a stem that extends from the body through pressurization of a lumen that is inflated by an internal muscular bladder called the ampulla (Fig. 1A,B). The stem and ampulla are each lined by a single muscle layer – circular in the ampulla and longitudinal in the stem – and the fluid exchange between the chambers allows these muscles to function as antagonists. This ability to generate extension depends on the helical winding around the stem. If initially retracted, a relatively high initial helix angle reduces toward the value that maximizes the internal volume (54.7 deg; Fig. 1E). In support of this idea, the tube foot in one species of sea star has been shown to possess a helix angle of 67 deg at rest (McCurley and Kier, 1995). Engineers have exploited the converse case to develop a pneumatic muscle-inspired motor called a McKibben actuator (Chou and Hannaford, 1996; Liu and Rahn, 2003; Tondu, 2012). The flexible walls of a McKibben actuator feature stiff fibers with a helix angle that begins at less than 54.7 deg, but increases during inflation to drive a contraction of the actuator's length.

In the present study, we developed an analytical framework for the transmission of mechanical work by soft skeletons. Our particular aims were to resolve whether variable gearing is common among a variety of hydrostatic systems and to determine the conditions under which MA is inversely related to DA. To this end, we modeled the geometry and mechanics in different hydrostatic skeletons to illustrate the effects of body deformation and helical winding on gearing. These models (summarized in Table 1) are of (1) a hydraulic press, (2) a McKibben actuator, (3) an echinoderm tube foot, (4) a cylindrical hydrostat without helical fibers, and (5) a cylindrical hydrostat with helical fibers. The first three of these are hydraulic systems where a fluid volume is exchanged between two chambers and the last two consist of a single chamber of fixed volume. The hydraulic press and McKibben actuator represent well-studied mechanical systems to which we can compare models of biological hydrostats.

Therefore, MA­_tf=1/DA_tf, as expected for a system without energy loss or storage. As a tube foot extends, helix angle will decrease and a maximum extension will occur at θ=54.7 deg, at which point mechanical advantage becomes zero and no output force is generated. Longer tube foot lengths will only occur if the tube foot sucker is pulled by an external force to lengthen the tube foot further.

DA_tf, MA­_tf and l­_tf (Eqns 24, 26 and 4, respectively) may be plotted parametrically with respect to c_amp (Eqn 23) over the range of helix angles that correspond to elongation (θ_mv<θ<90 deg, Fig. 3). As the circumference decreases, the tube foot lengthens and helix angle decreases (Fig. 3B). As in the piston-plus-McKibben compound machine, these patterns result in an elevation of DA_tf and reduction of MA­_tf near full extension of the tube foot (Fig. 3C). However, the gearing of the piston-plus-McKibben actuator shows a monotonic decrease in mechanical advantage (Fig. 2H) as it lengthens, whereas the tube foot has a maximum mechanical advantage that occurs at some intermediate length as the stem extends (Fig. 3C). That maximum occurs at a circumference that depends on the relative size of the ampulla and tube foot stem.

We illustrate the properties of the tube foot with the internal spring using parametric plots of the relevant equations over the same range of helix angles as previously. While stem length (l_tf) and displacement advantage (DA_tf) are independent of the effect of muscle spring constant (Fig. 4B), mechanical advantage (MA_tf,spring) changes with the spring constant (Fig. 4C; Eqn 29). Consequently, once the spring engages, transmission efficiency (i.e. MA_tf,spring×DA_tf; Fig. 4D) varies as the stem extends. This is an example where energy storage in an elastic element results in η<1.

The addition of an effective spring reduces the maximum elongation of the tube foot achievable through ampullar contraction, presenting an intriguing mechanism to control output displacement of McKibben-type systems. As a McKibben actuator without an internal longitudinal spring approaches its maximum volume, displacement advantage diverges rapidly, meaning that even a small change in input length will correspond to a large change in output displacement. A sufficiently stiff spring could reduce the operating length of the McKibben actuator such that displacement advantage is nearly linear over the full range, which may allow more precise control of output displacement.

The cylindrical hydrostat includes a constant volume of incompressible fluid or tissue volume lined with longitudinal and circular muscles. Fiberless models have been previously employed to consider the displacement advantage of a squid's tentacle (Kier, 1982; Kier and Smith, 1985; Kier and Van Leeuwen, 1997) and the fluid-filled body of an earthworm (Kurth and Kier, 2014, 2015). In these models, either transverse and circular muscles (e.g. the squid) or circular muscles alone (e.g. the earthworm) cause axial extension of the body (Fig. 5A). Any differences in the gearing between circular and transverse muscles would help to indicate the functional implications of their presence and the circumstances where it may be beneficial for their activation on extension. The longitudinal muscles reverse this extension and apply a radial force by pressurizing the body's interior and causing radial expansion. This radial force may act on the surrounding environment (Fig. 5D), which permits an earthworm to anchor some segments against a burrow (Quillin, 2000). Here, we neglect any internal resistance from stretching skin and hence the radially-directed forces act entirely on the environment. As we demonstrate, the mechanical advantage and displacement advantage depend on whether circular or transverse muscles are generating the extension.

When comparing mechanical and displacement advantage between extension and retraction, it is crucial to identify muscle orientation, relevant input and output directions and distances over which work is done (Fig. 5).

Cylindrical hydrostats generally possess a cross-helical winding composed of collagen fibers (Clark and Cowey, 1958;Wainwright, 1988; Shadwick, 2008). The helix angle of these fibers is approximately 54.7 deg in the body wall of a diversity of worms (Clark and Cowey, 1958; Kier, 2012), some polychaetes (Law et al., 2014), and in the notochords of developing vertebrates (Adams et al., 1990; Koehl et al., 2000). As for hydraulic systems, one mechanical role of helical fibers in hydrostats is thought to be to aid in maintaining its cylindrical shape as it deforms (Wainwright, 1988). However, forces transmitted by these fibers have not previously been explicitly included in models of helically wound cylindrical hydrostats. Here, we consider the geometry and mechanics of fiber winding around a constant-volume cylindrical hydrostat and we develop expressions for mechanical advantage, displacement advantage and transmission efficiency.

A pressurized helically wrapped, constant-volume cylinder will have the fibers at an angle of 54.7 deg in the absence of external forces, and a change in shape will tend to reduce the volume (Fig. 1E). Therefore, for an incompressible object where volume cannot change, fibers must extend. For example, the body segment of an earthworm that is filled with a volume of fluid accommodated by a helix angle of 54.7 deg requires that the fiber changes length for the segment to change in length.

The action of circular muscles serves to modulate body mechanics. Contraction of circular muscles generates an axial pushing force that is higher at a given fiber angle than it was at that fiber angle when circular muscles were not activated (Fig. 6E). The circumference (and hence helix angle) at which zero axial force is achieved depends on the stress generated by circular muscles (inset in Fig. 6E). If the body were to be stretched beyond this zero axial force point, a pulling force would tend to return the body to its balanced state. This restorative force is principally due to the passive mechanics of the fibers as they become more nearly aligned with the axial direction, and hence become more independent of the level of circular muscle activation.

At all muscle stresses, a pressure minimum occurs at 54.7 deg, which is sometimes called the ‘magic angle’ (Horgan and Murphy, 2018, 2022); this the angle where loads in the fibers balance the hoop and longitudinal tension in the cylinder due to pressure loads. Hoses are typically manufactured with helical reinforcing fibers at this angle to prevent sudden elongation or shortening when the hose is pressurized. It can be shown that 54.7 deg is the pressurization magic angle for axial extension under the assumptions of a thin membrane and inextensible fibers (Demirkoparan and Pence, 2015). Our equations show that in constant-volume cylindrical hydrostats with extensible fibers, circular muscle stresses can carry some of the pressure load that would otherwise be carried by the helical fibers and that this load sharing causes the magic angle to move to lower angles corresponding to longer, thinner cylinders.

Consider a given cylinder (constant n, E) with a specified σ_m. As this cylinder changes shape, concomitant to changes in fiber angle, the mechanical and displacement advantage in retraction and extension (Eqns 57–62) can be plotted (Fig. 7). Generally, displacement advantage decreases and mechanical advantage and transmission efficiency increase as 54.7 deg is approached. Transmission efficiency is equal to unity only when the helix angle is 54.7 deg, where no energy is stored in the helical fibers. There are many cases where this may be valid because a variety of biological systems maintain a fiber winding close to 54.7 deg.

However, both extension and retraction of the body stretch the helical fibers, store elastic energy and thereby reduce transmission efficiency as the body deforms (Fig. 7E). Factors affecting mechanical advantage can be identified from Eqns 57 or 60. Energy storage will be greater further from 54.7 deg and is affected by both circular muscle stress and the stiffness of the helical fibers. Specifically, the mechanical advantage curve will be less than the inverse of displacement advantage to a greater extent when the fiber stiffness creates resistance that is substantially greater than the force of the circular muscles.

Our mathematical models consider the transmission of force and displacement by a variety of hydrostatic skeletons. The principal aims for this modeling were to examine how variable gearing emerges among hydrostatic skeletons and to evaluate the conditions where the mechanical advantage is equivalent to the inverse of the displacement advantage (Table 1). A prevailing theme in the predictions of these models is that hydrostatic skeletons transmit force with variable gearing as the structure deforms (Figs 2H, 3C, 4B,C, 5C,F, and 7D). Our modeling also reveals the importance of identifying input force and displacement axes, and the contribution of energy storage and dissipation to this variable gearing. Incorporating these considerations into calculations of mechanical advantage, displacement advantage and transmission efficiency informs our understanding of biological soft skeletons and the design of engineered devices. While the present results offer an analytical framework, the utility of our models remains to be tested experimentally.

We found that variable gearing emerges in hydrostatic skeletons in different forms. The hydraulic press exhibits no variable gearing: the rigid walls of its two chambers yield constant mechanical and displacement advantage (Fig. 2A–C, Eqns 9 and 10). The hydraulic press consequently operates like a rigid joint (Fig. 1F), despite transmitting pressure through a liquid medium. However, replacing one of these chambers with the deformable McKibben actuator introduces variable gearing as a function of the helical winding (Eqns 12 and 21, Fig. 2D–H). The rapid, vertically asymptotic increase in the displacement advantage of the McKibben actuator (Fig. 2H) may appear superficially similar to that of a cylindrical hydrostat (Fig. 5), as previously proposed (Kier and Smith, 1985), but is different because it occurs towards the magic angle (54.7 deg), in the middle of the range of possible cylinder shapes. The special condition in the middle of the shape range was previously unrecognized in the biological literature and applies to all helically wrapped pressurized structures considered (Figs 5 and 7).

The tube foot offers perhaps the most intriguing form of variable gearing. The combination of the ampullar and fiber winding geometries of the stem result in a maximum mechanical advantage in the middle of stem extension (Fig. 3C). This is unlike the monotonic decline in mechanical advantage of the McKibben actuator (Fig. 2H) and cylindrical hydrostat (Fig. 5C). Thus, the relative dimensions of the ampulla and tube feet could potentially serve to maximize force transmission at a length that corresponds to the power strokes that drive the bouncing gait of sea stars (Ellers et al., 2014, 2021; Heydari et al., 2020). Additionally, perhaps the stem length that maximizes force transmission corresponds to the maximum force in the length–tension relationship for the muscles in the wall of the ampulla. If so, the output force could be maximized through a combination of muscle tension and skeletal force transmission.

Another previously unrecognized phenomenon is that gearing depends on the direction of muscular input. For a constant-volume circular hydrostat, we define the input displacement by circular muscles as a change in circumference, which contrasts with the previous convention of modeling changes in radius or diameter (Quillin, 1999; 2000; Kier, 2012; Law et al., 2014; Kurth and Kier, 2014, 2015). This distinction changes the anticipated displacement and mechanical advantage. For example, the displacement advantage of a circular muscle in a fixed-volume extending cylindrical hydrostat is less than that of a transverse muscle by a factor of π. Consequently, transverse muscles in squid tentacles contribute to speedy extension about 3 times more than would circular muscles.

Our fiber-wound models show that the helical fibers create a reference point at 54.7 deg relative to which predictions about mechanical and displacement advantage can be made. For instance, our model of a constant-volume helically wrapped hydrostat shows that for a specific set of parameters, the mechanical advantage is higher when retraction generates radial output forces just above 54.7 deg than when extension generates axial output forces just below 54.7 deg (Fig. 7D). The relative size of the mechanical advantage close to 54.7 deg is however set by the ratio of length and radius (Eqn 65). Our model has reset the perspective regarding MA and DA from one focused primarily on the length to diameter ratio to one focused on the fiber angle and its interaction with muscular forces. Just as in the McKibben actuator, in which the structure can only be understood with respect to the fiber angle, and because the McKibben actuator operates in two modes (extending or retracting towards the 54.7 deg length) a constant-volume cylindrical hydrostat's mechanical behavior must be analyzed relative to the fiber angle. Further models with time-sequenced applications of longitudinal and circumferential muscle forces, muscle properties and assumed external forces must be developed to fully utilize our worm model.

Energy relates mechanical and displacement advantage. If output work (l_outF_out) generated by a skeleton is equal to its input work (l_inF_in), then MA and DA are inverses, which corresponds to an ideal transmission efficiency (Eqn 3). Therefore, geometric modeling offers a comprehensive understanding of the transmission of mechanical work through a hydrostatic skeleton when energy dissipation or storage is negligible. This insight applies also to rigid skeletons.

There are two ways that a skeleton may fail to transmit all input work to the system's output. First, energy may be dissipated as the structure deforms. For example, tube foot inflation requires fluid flow through small channels that may incur viscous losses. Similar dissipation from hydrodynamic drag has been modeled as a major factor in the force transmission of the raptorial appendage of stomatopods, because of its rapid motion (McHenry et al., 2012, 2016). Second, some of the input energy may be stored elastically. We have considered this through examples of the extension of the longitudinal muscles in the stem of the tube foot (Fig. 4) and the extension of helical fibers in the body wall of a worm (Fig. 7). The degree to which such storage occurs must be determined with experiments. For example, a preparation where the lumen of the ampulla is pressurized to a controlled extent and the muscles are rendered inactive would provide the opportunity to control the input force within a tube foot. Measurement of the output force and geometry of the tube foot would allow for measurement of the mechanical advantage, which could be compared with the predictions of our model to evaluate the extent of energy storage and dissipation in the tube foot system.

Our consideration of energetics both compliments and contrasts with previous attempts to determine the mechanical advantage by inference from the displacement advantage (Kier and Smith, 1985; Quillin, 1999; Kurth and Kier, 2014, 2015). Kier and Smith (1985) used an analogy to lever systems when they proposed that a high displacement advantage in cylindrical fixed-volume hydrostats should yield a low mechanical advantage, an approach that was adopted in other considerations of soft skeletons (Wainwright, 1988; Quillin, 2000; Vogel, 2013). For earthworms, Kurth and Kier (2014, 2015) explicitly assumed MA as the inverse of DA, with the displacement advantage calculated from morphometrics. By modeling the mechanics of a range of hydrostatic skeletons, we have shown how the mechanical advantage may be calculated as the inverse of the displacement advantage in a variety of systems, but only if energy is neither stored nor dissipated by the system.

The variety of helically wrapped structures considered here offers the opportunity to expand on the traditional presentation of geometric models. Classic work (Clark and Cowey, 1958) emphasized how the volume enclosed by a cylinder (Eqn 7) varies with the helix angle and the graphical presentation of this relationship has served as the centerpiece of textbook introductions to soft skeletons (Wainwright, 1976; Alexander, 1983; Vogel, 2013). However, this presentation can be difficult to reconcile with a cylindrical hydrostat that has a fixed volume and extensible fibers. A fixed volume requires that the system remain at one level along the ordinate. An extension of the fibers graphically shifts the operating point towards a longer or shorter cylinder when helix angle is less than or greater than 54.7 deg (blue arrows in Fig. 8A). Shortening of the fibers has the opposite effect on cylinder length, moving the cylinder length towards the 54.7 deg length. An alternative graphical presentation shows the fiber length and helix angle varying along the contours of a fixed volume (Fig. 8B). Hence, shape change, fiber length change and helix angle covary (Fig. 8C). This new representation clarifies changes in a deforming hydrostat.

Regardless of the presentation, the current work compliments previous classic geometric models of fixed-volume cylindrical bodies (Cowey, 1952; Clark and Cowey, 1958; Kier and Smith, 1985; Kier, 2012). Our derivations similarly show that the same displacement advantage is predicted for a constant-volume hydrostat with (Eqn 58) or without helical fibers (Eqn 32). In contrast, the mechanical advantage is additionally dependent on energy storage in helical fibers. Moreover, when volume is not constant, both mechanical and displacement advantage depend on the fiber angle, as illustrated by our analysis of the McKibben actuator. Thus, our analysis considerably extends the previous understanding of these systems by explicitly modeling their mechanics.

Our study deviates from prior work on soft skeletons in our explicit consideration of forces. Our models relate internal pressure to stress in helically wrapped fibers and reveal that a proportion of input work is stored as elastic energy. Such storage causes calculated MA to diverge from the inverse of DA, thus rendering potentially inoperable the widespread previous practice of using DA to estimate MA. The degree to which storage impacts force transmission will need to be measured experimentally. Our models should also enable the investigation of fiber wrapping diversity. For example, a peristaltically burrowing polychaete worm in the family Opheliidae (Law et al., 2014) has anterior circular muscles and a cuticle fiber helix angle of 45 deg, whereas an undulatory burrower has no anterior circular muscles and a cuticle fiber helix angle of 54.7 deg. Nematodes, which have their own distinctive motions, are pressurized, cylindrical and have reported fiber angles of 75 deg (Harris and Crofton, 1957) or 54.7 deg (Gans and Burr, 1994). Further fiber-wound systems are caecilians, muscular hydrostats, fish bodies and connective-tissue wrapped muscle. Our conceptualizations suggest a reanalysis of these systems is warranted.

It has long been recognized that fibers necessarily change length and helix angle as a constant-volume cylindrical body changes shape (Clark and Cowey, 1958), and further that fiber length is a minimum at 54.7 deg (eqn 5 of Kier and Smith, 1985, equivalent to our Eqn 38 for n=1). From this observation, it was surmised that contracting helical muscle in a squid tentacle would cause elongating or shortening forces when the helix angle was greater or less than 54.7 deg, respectively. This corresponds to our case of zero circular muscle tension in a cylindrical hydrostat (violet line in Fig. 6D). However, we additionally modeled the patterns of axial forces when the circular muscles generate force. When circular muscles generate tension, there is a change in the critical fiber angle (and thus cylinder circumference) at which axial forces switch from pushing to pulling. Instead of occurring at 54.7 deg, the switch occurs at the fiber angle corresponding to the circumference where axial force is zero (inset of Fig. 6E). In short, axial pushing forces are generated by tension in helical fibers even at fiber angles less than 54.7 deg, when the circular muscles contract.

Muscular forces interact with the mechanics of a hydrostatic skeleton to influence the internal fluid pressure. This has been shown in experimental measurements of pressure and muscle contraction in earthworms (Seymour, 1969; Quillin, 1998; 2000). Circular muscle contraction caused pressure increases during elongation and the subsequent longitudinal contraction caused pressure decreases during shortening. This is consistent with the prediction of pressure in the region just to the right of 54.7 deg in Fig. 6D. One of the interesting consequences of the helical fibers generating forces opposing lengthening is that when there are no external axial forces acting, the length of the worm is restricted to the length that corresponds to the force balance between the helical fibers and the pressure in the coelom, resulting in the magic angle being modulated by the hoop muscle stress. This can be seen on Fig. 6E, where the axial force curves cross zero. The length of a helically wrapped worm is constrained by its helical winding.

Squid strike prey through the rapid extension of their tentacles. These hydrostatic structures accelerate and decelerate within a few milliseconds, but are also capable of a long extension (70%; Kier, 1982; Kier and Van Leeuwen, 1997). The initial dimensions of the tentacles (20:1 length:diameter ratio; Kier and Smith, 1985) should favor a relatively modest mechanical advantage (∼0.08, for θ=54.7 deg) that declines over the course of a strike (Fig. 5C). However, these dimensions are appropriate for a high displacement advantage, which will tend to increase during the strike, provided a negligible role of fiber winding or any other connective tissue within the tentacle. If the helical fibers possess substantial stiffness, then the fiber angle changes should be quite high and thereby store elastic energy as the tentacles nearly double in length. Alternatively, fiber and muscle angles could be arranged to store energy like a spring before the strike or they may be highly compliant and thereby play little role in the tentacle's dynamics.

The role of connective tissue in the mechanics of a hydrostatic skeleton may be investigated experimentally. For example, for squid tentacles, our models suggest different modes of functioning that may be differentiated by measurements of internal pressure and electromyography. Our results suggest that preloading of helical fibers or relatively compliant fibers is necessary for a pressurized cylinder to deform to the extent exhibited by the tentacles. A forward dynamics model (Van Leeuwen and Kier, 1997) suggests that the transverse muscle arrangement with short sarcomeres is sufficient to explain the rapid strike, with a negligible role from helical fibers. Indeed, our calculations also suggest that the use of transverse rather than circular muscle increases the displacement advantage by a factor of π. The ultrastructure of the tentacles largely explains how the transverse muscles can achieve a maximum strain rate that is an order of magnitude faster than that of the transverse muscles of the squid arms (Kier and Curtin, 2002), but it remains unclear whether fiber winding or tendons store a significant level of elastic energy over the course of a strike.

Our modeling of echinoderm tube feet reveals a number of mechanical principles. We found that the mechanical advantage declines to zero as the helix angle approaches 54.7 deg (Fig. 3C) and that the stem can only extend further if pulled by an external force. At higher values of the helix angle, a maximum mechanical advantage occurs at an intermediate length, which presents intriguing possibilities for maximizing force output. The transmission efficiency declines when a longitudinal muscle spring is incorporated into our model because of elastic energy storage (Fig. 4D), and a high stiffness of this spring lowers the mechanical advantage and reduces the maximum elongation possible. However, the existence of the spring could aid in the control of extension by resisting the rapid increase in the displacement advantage near θ≈54 deg. These results are rich with possibilities for experimental tests of the system that could monitor or manipulate internal pressure while measuring the deformation of these structures.

The geometry of worm bodies determines whether they are functioning with high displacement advantage or high mechanical advantage when engaged in common behaviors such as extension and retraction. Several experimental studies have documented pressures and forces involved but have not incorporated consideration of the transmission and likely storage of mechanical energy in the helical fibers despite having documented their presence. Redesign of those experiments to understand the role of the helical fibers is warranted. In particular, our equations make specific predictions of how mechanical advantage will be affected as helix angle changes. Measuring helical fiber mechanical properties and tracking helix angle and pressure during force experiments similar to those of Quillin (2000) would test whether the fibers play a role in the transmission of force. Alternatively, the fibers may play a negligible role if the angle stays close to 54.7 deg or if the fibers are compliant.

Our insights into the force-transmission properties of soft skeletons open opportunities for comparisons among animals and between natural and robotic devices. In a manner analogous to the study of the functional anatomy of vertebrates (Smith and Savage, 1956; Alexander, 1983), our definitions for mechanical advantage may be applied to functional morphology of soft structures in invertebrates. Our definitions also allow mechanical comparisons among hydrostatic skeletal types and between soft and jointed skeletons. Our concepts may also aid in the design of soft robotic devices to optimize force and displacement transmission for a variety of applications.

This paper was largely inspired by the work of W. M. Kier and benefited from conversations with him and C. Rahn. Three anonymous reviewers provided highly valuable feedback on the initial version of the manuscript.