A hydrostatic skeleton allows a soft body to transmit muscular force via internal pressure. A human's tongue, an octopus' arm and a nematode's body illustrate the pervasive presence of hydrostatic skeletons among animals, which has inspired the design of soft engineered actuators. However, there is a need for a theoretical basis for understanding how hydrostatic skeletons apply mechanical work. We therefore modeled the shape change and mechanics of natural and engineered hydrostatic skeletons to determine their mechanical advantage (MA) and displacement advantage (DA). These models apply to a variety of biological structures, but we explicitly consider the tube feet of a sea star and the body segments of an earthworm, and contrast them with a hydraulic press and a McKibben actuator. A helical winding of stiff, elastic fibers around these soft actuators plays a critical role in their mechanics by maintaining a cylindrical shape, distributing forces throughout the structure and storing elastic energy. In contrast to a single-joint lever system, soft hydrostats exhibit variable gearing with changes in MA generated by deformation in the skeleton. We found that this gearing is affected by the transmission efficiency of mechanical work (MA×DA) or, equivalently, the ratio of output to input work. The transmission efficiency changes with the capacity to store elastic energy within helically wrapped fibers or associated musculature. This modeling offers a conceptual basis for understanding the relationship between the morphology of hydrostatic skeletons and their mechanical performance.

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.

List of symbols and abbreviations

     
  • a

    transverse cross-sectional area

  •  
  • A

    area for force transmission

  •  
  • c

    circumference of cylinder

  •  
  • DA

    displacement advantage

  •  
  • E

    Young's modulus

  •  
  • f

    length of fiber for single rotation of helix

  •  
  • F

    force

  •  
  • k

    spring stiffness

  •  
  • l

    cylinder length

  •  
  • MA

    mechanical advantage

  •  
  • n

    number of rotations of the helix on the cylinder

  •  
  • P

    pressure

  •  
  • v

    volume of body wall

  •  
  • V

    volume of fluid in a chamber

  •  
  • V

    volume of constant-volume cylinder

  •  
  • r

    cylinder radius

  •  
  • t

    wall thickness

  •  
  • T

    tension

  •  
  • W

    work

  •  
  • strain

  •  
  • η

    transmission efficiency, MA×DA

  •  
  • θ

    helix angle

  •  
  • κ

    ratio of wall volume to cylinder volume

  •  
  • σ

    stress

  •  
  • Subscripts

     

  •  
  • 54

    cvf with fibers at θ=54.7 deg

  •  
  • amp

    ampulla

  •  
  • axial

    along the longitudinal axis

  •  
  • circ

    circular muscles

  •  
  • ­cv

    constant-volume cylindrical hydrostat

  •  
  • cvf

    cv with helically wrapped fibers

  •  
  • extension

  •  
  • E=0

    cvf with fibers at zero stiffness

  •  
  • f

    fiber

  •  
  • h

    hoop

  •  
  • in

    input

  •  
  • l

    longitudinal

  •  
  • long

    longitudinal muscles

  •  
  • m

    muscle

  •  
  • Mn

    piston-plus-McKibben actuator

  •  
  • mv

    maximum volume

  •  
  • o

    equilibrium or resting value of a quantity

  •  
  • out

    output

  •  
  • Pi

    piston of piston-plus-McKibben

  •  
  • press

    hydraulic press

  •  
  • retraction

  •  
  • spring

    spring

  •  
  • tf

    tube foot

  •  
  • tot

    total

  •  
  • w

    wall

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.

DA and MA in hydrostatic skeletons

The geometry and mechanical properties of a skeleton determine how it will transmit input force and displacement. The skeleton is actuated by an input force (Fin), that is exerted over some displacement (dlin) in the same direction to apply work (dWin) to the system. This work is transmitted through the skeleton to its output end, where force (Fout) is applied over a displacement (dlout) in the same direction to generate output work (dWout). The amount of output displacement generated for a given input is determined by the DA, which is equal to the magnitude of the following derivative:
formula
(1)
The choice of input and output directions is important in this calculation and should reflect the input and output work that is relevant to the situation. For instance, when an earthworm segment retracts, its input force will be generated by longitudinal muscles and its displacement would be the axial distance through which the muscles contract. Less obvious is the case of the input displacement for the segment's circular muscles, where the input displacement is generated by a reduction in the circumference (i.e. dlin=dc=2πdrin). This contrasts with a reduction in the radial direction that may be directly generated by contraction of transverse muscles (i.e. dlin=drin) in some muscular hydrostats.
The mechanical advantage also varies with the geometry of a system, but is defined as the ratio of the magnitudes of output and input forces:
formula
(2)
If no energy is stored or dissipated in the structure, then the output work and the input work are equal, and thus:
formula
(3)
Therefore, the product of MA and DA provides a measure of the transmission efficiency η of mechanical work through a skeleton. A transmission efficiency of less than unity indicates either dissipation of mechanical energy from the system or storage in the form of elastic energy in the skeleton.

The geometry of helical fibers

Cross-helical stiff fibers transmit forces and play an important role in determining the mechanical and displacement advantage in a variety of soft skeletons. As in previous studies (Cowey, 1952; Clark and Cowey, 1958; Kier and Smith, 1985;Wainwright, 1988; Koehl et al., 2000; Kier, 2012), our models consider fibers wrapped helically around a cylinder at a helix angle θ from the longitudinal axis of the cylinder (Fig. 1C). In some cases below, we assume inextensible (i.e. fixed-length) fibers enclosing a variable volume (e.g. the McKibben actuator and tube foot). In other models (e.g. the cylindrical hydrostat with helical winding), the volume is constant while the fiber length varies. In either case, we assume f is the length of a fiber through a single rotation of the helix (Fig. 1D) and n is the number of rotations of the helix on the cylinder. Then, the cylinder enclosed by fibers at an angle θ has length l, radius r and transverse cross-sectional area A given by:
formula
(4)
formula
(5)
formula
(6)
The enclosed cylinder's volume using Vlr2 with Eqns 4 and 5 is:
formula
(7)
For the inextensible fiber case, plots of these variables (Fig. 1E) as shape changes, show the maximum volume occurring at θmv=arctan√2≈54.7 deg. This fact was used in the first interpretation of the deformation of nemertean worms (Cowey, 1952) and generalized by Clark and Cowey (1958) for a broader diversity of worms.
The rate of change of length (Eqn 4) with respect to volume (Eqn 7) is:
formula
(8)
and reveals that length decreases if 0<θ<θ­mv or increases if θ­mv<θ<π/2, as the volume approaches the maximum (Fig. 1E).

The hydraulic press

The properties of a hydraulic press are well established (Miller et al., 2004), but we repeat them here for contrast with systems with deformable walls. The hydraulic press consists of rigid cylindrical input and output chambers filled with a constant volume of an incompressible fluid that is free to move between them. This fluid transmits forces from the input to the output pistons (Fig. 2A). The properties of the hydraulic press are determined by the geometry of the chambers. Because of their rigid walls, the transverse cross-sectional area of each chamber (Ain, A­out, respectively) is fixed and therefore the displacement of the two pistons (dlin, dlout) is related by the constant-volume condition. This yields the following displacement advantage (Eqn 1) for the hydraulic press:
formula
formula
(9)
Assuming equal pressure P in the chambers, the mechanical advantage (Eqn 2) for the hydraulic press is:
formula
(10)
Thus, transmission efficiency for the hydraulic press is:
formula
(11)
where Wout and Win are work-out and work-in, respectively. The inverse relationship between MApress and DApress is a manifestation of the complete transmission of energy from the input to the output chambers. In addition, the displacement of the output piston is directly proportional to that of the input piston (Fig. 2A) and hence MApress (Eqn 10) and DApress (Eqn 9) are fixed across all positional changes (Fig. 2C), as in a lever (Fig. 1F).

The McKibben actuator

The McKibben actuator consists of a deformable fluid chamber constrained by inextensible, cross-helical fibers (Fig. 2D). We modeled the McKibben actuator as inflated by the piston of a rigid input chamber with both chambers filled by an incompressible fluid. We assumed that the volume leaving the piston chamber is the same as the volume entering the deformable chamber and used the rate of change in length with respect to volume (Eqn 8) to calculate DAMn for the piston-plus-McKibben actuator:
formula
formula
(12)
In contrast to the hydraulic press, where DApress is constant as the piston pushes (Eqn 9), DAMn changes as the McKibben actuator changes volume and length, which alters gearing. For an elongating McKibben actuator (i.e. θmv<θ<90 deg), length (Eqn 4) increases until the helix accommodates the maximum volume at θ=θ­mv. Thus, DAMn (Eqn 12) increases rapidly near to that maximum-volume full length (Fig. 2H).
To determine mechanical advantage of the piston-plus-McKibben actuator, MAMn, we first calculate axial output force. We derive this force, previously derived using an energy argument (Chou and Hannaford, 1996), by considering stress in the helical fibers. We assume inextensible fibers that carry all forces in tension, with no contribution from any other material in the cylinder wall. We relate stress in the helical fibers σf to hoop stress σh and longitudinal stress σl in the cylinder wall (Fig. 2F):
formula
(13)
formula
(14)
Consider a force balance across a longitudinal plane of the cylinder with wall thickness t:
formula
(15)
Using Eqn 13 in Eqn 15, stress in the helical fibers is:
formula
(16)
The longitudinal output force is the sum of the force due to pressure on the end of the cylinder, minus the horizontal component of the force exerted from the helical fibers in the wall on the cylinder ends:
formula
(17)
with area of the transverse section of the cylinder wall being:
formula
(18)
which assumes a thin wall as in Demirkoparan and Pence (2015). The output force for the piston-plus-McKibben system, using Eqns 5, 6, 14, 16 and 18 with Eqn 17 is:
formula
(19)
This relationship shows that zero force is generated at θmv, negative (i.e. pulling) forces are generated where 0 deg<θ<θmv, and positive (i.e. pushing) forces occur where θ­mv<θ<90 deg. Note that our sign convention for forces is the opposite of Chou and Hannaford (1996).
The input force on the piston and the internal pressure in the machine are related:
formula
(20)
Mechanical advantage for the piston-plus-McKibben compound machine (using Eqn 19) is thus:
formula
(21)
Therefore, by comparison with its displacement advantage (Eqn 12), MAMn=1/DA­Mn and mechanical advantage hence declines towards zero as displacement advantage increases near the full extension of the actuator. This inverse relationship indicates an ideal transmission efficiency (η=1) and complete conversion of input work to output work by the actuator.
We used parametric plots to illustrate how the piston-plus-McKibben compound machine exhibits variable gearing as it is inflated (Fig. 2G,H). We evaluated DAMn, MA­Mn and lout (Eqns 12, 21 and 4, respectively) over the range of helix angles that correspond to elongation of the actuator (90 deg>θ>θ­mv). These quantities are plotted against displacement of the piston (lin) as a function of helix angle using the assumptions of constant volume, fluid incompressibility and rigid input-chamber geometry:
formula
(22)
These plots illustrate that the actuator length increases more rapidly with input displacement at greater extension (Fig. 2G). This trend results in an increase in DAMn and a corresponding decrease in MAMn as the actuator extends (Fig. 2H). Therefore, the McKibben actuator is most effective at transmitting force at low levels of inflation and is geared more for displacement when inflated close to its maximum volume.

The tube foot

We modeled the tube foot as a two-chambered system consisting of ampulla and stem. The stem of the tube foot is a fiber-wound cylindrical structure (Figs 1B and 3A) that we modeled like the McKibben actuator (Fig. 2D–H). We modeled the compliant ampulla as a cylinder with a fixed length (lamp) and variable circumference (camp), consistent with previous work (Kier, 2012). The compliant ampulla distinguishes the tube foot from the piston-plus-McKibben actuator. However, we similarly relate volumetric changes of input and output chambers. Specifically, assuming a constant-volume sum of ampulla (here Vin) and tube foot (Vout, Eqn 7), the circumference of the ampulla, as a function of helix angle, is:
formula
formula
formula
(23)
Circumference is the relevant dimension of the input chamber when calculating the displacement advantage because work is generated by contraction of circumferential muscles. Displacement advantage (Eqn 1) is the rate of stem extension with respect to changes in the ampulla's circumference (Eqns 8 and 23):
formula
(24)
The compliant ampulla additionally plays a role in the mechanical advantage. Circular muscles of the ampulla generate an input force equal to the product of tension T and the length of the ampulla (Fin=Tlamp). Using Laplace's Law for hoop tension in a cylinder, one may relate this muscular tension to the circumference of the ampulla (P=2πT/camp). Therefore, pressure generated by the ampulla is:
formula
(25)
Next, we determined the mechanical advantage in two cases: with and without energy storage by elastic elements in the structure.

Without elastic energy storage

We assumed that all input work is transmitted to the output work. Combining our models for the output force (Eqn 19) and pressure (Eqn 25), gives the mechanical advantage for the tube foot MAtf:
formula
formula
(26)

Therefore, MA­tf=1/DAtf, 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.

DAtf, MA­tf and l­tf (Eqns 24, 26 and 4, respectively) may be plotted parametrically with respect to camp (Eqn 23) over the range of helix angles that correspond to elongation (θmv<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 DAtf and reduction of MA­tf 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.

With elastic energy storage

Here, we consider the effects of energy storage due to passive force from longitudinal muscles in the stem (Fig. 1B). We speculated that the longitudinal muscles are initially folded when the stem is retracted and after reaching the muscles' unfolded length, and start generating a passive linearly elastic tensile force in proportion to their length beyond that point. This force counters the extension of the tube foot beyond the unfolded length, but offers no resistance at shorter lengths. This spring-like force is:
formula
(27)
where ktf is the stiffness of the tube foot spring (Fig. 4A). This spring approximates the passive elastic component of the longitudinal muscles and not their active tension. The spring resting angle θout,o, as a function of ltf,o, can be calculated from the length of the helix (Eqn 4):
formula
(28)
Accounting for force exerted by this spring (Eqn 27 in Eqn 26) yields a modified mechanical advantage:
formula
(29)
This equation for the mechanical advantage deviates from the inverse of DAtf (Eqn 24) for any non-zero spring stiffness, when the tube foot extends beyond the resting length of the spring. This is because the passive elasticity of the longitudinal muscle stores some of the energy generated by the ampulla, rather than allowing all work to be transmitted to extending the tube foot. Once the ampulla relaxes, that stored elastic energy may then contribute to passive shortening of the tube foot. However, if the resistance to lengthening were instead a consequence of damping by the longitudinal muscle (not presently modeled), then the energy dissipation would reduce the output force but not contribute to subsequent retraction.

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 (ltf) and displacement advantage (DAtf) are independent of the effect of muscle spring constant (Fig. 4B), mechanical advantage (MAtf,spring) changes with the spring constant (Fig. 4C; Eqn 29). Consequently, once the spring engages, transmission efficiency (i.e. MAtf,spring×DAtf; 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 fiberless cylindrical hydrostat

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.

The displacement advantage of a cylindrical hydrostat derives from its length l and radius r:
formula
(30)
formula
(31)
where and c are, respectively, the volume and circumference of the constant-volume cylinder. The displacement advantage for the extension of a cylindrical body by the contraction of circular muscles is the derivative of length with respect to circumference:
formula
(32)
This length to radius ratio increases rapidly as the body becomes more elongated (Fig. 5B,C). If the muscles doing work were transverse, then the relevant input length would be diameter d instead of circumference, in which case we would take the derivative of length with respect to diameter, which gives a displacement advantage that is a factor of dc/dd=π larger than that in Eqn 32. Thus, the transverse muscles of a squid tentacle offer substantially greater amplification of displacement during extension, and correspondingly less mechanical advantage, than would circular muscles.
We now consider the mechanical advantage for extension based on the contraction of circular muscles. This force is generated by hoop tension, Th, which is related to the pressure in the cylinder P by Laplace's Law:
formula
(33)
Thus, the input force is:
formula
(34)
The axial force output at the cylinder's end is the product of pressure and circular cross-sectional area of the cylinder; therefore, the mechanical advantage is:
formula
(35)
Thus, (Eqn 32), as must be the case in the absence of energy dissipation or storage. If the muscles generating the input force were transverse instead of circular, then the mechanical advantage would be lower than that calculated in Eqn 35 by a factor of 1/π.
During retraction, activation of the longitudinal muscles serves to shorten the body and thereby increase its radius. Given the relationship of the radius to the volume of the body (Eqn 31), the displacement advantage is:
formula
(36)
Therefore, the displacement advantage increases rapidly as the longitudinal muscles contract (Fig. 5E,F).
The mechanical advantage for retraction considers the radially directed output force that is transmitted by pressure. The input force generating the pressure is created by the longitudinal muscles acting over the cylinder ends. The mechanical advantage in retraction is thus:
formula
(37)
Therefore, (Eqn 36) and MAℝ,cv declines during retraction (Fig. 5F).

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).

The cylindrical hydrostat with elastic cross-helical fibers

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.

Helical geometry with changes in fiber length

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 fiber length per turn of the helix is found by rearranging the equation for the volume enclosed by the helix (Eqn 7):
formula
(38)
The resting fiber length per rotation of the helix (fo) occurs at the equilibrium angle (θ­mv=54.7 deg) when there are no external longitudinal or radial forces, giving:
formula
(39)
The change in fiber length from this resting length to achieve a particular helix angle is the difference between the fiber length for that angle (Eqn 38) and the resting fiber length (Eqn 39). Given variable fiber length and constant volume, the cylinder's length, radius and cross-sectional area with respect to helix angle are calculated by substituting Eqn 38 into the equations for helical geometry (Eqns 4, 5 and 6):
formula
(40)
formula
(41)
formula
(42)
Note that, in all derivations, we assume that physical parameters are greater than zero and that fiber angle is 0<θ<π/2.
Next, we calculate the ratio of length to radius in the context of a fiber angle. This variable was important in calculating the mechanical and displacement advantage for cylindrical hydrostats without fibers, and we now want to be able to compare hydrostats with and without fibers. Using Eqns 40 and 41:
formula
(43)
Thus the shape of the cylinder depends on both the number of wrappings and the fiber angle. Consider the geometry for earthworms. Average length to diameter ratio for whole earthworms was 102/5.3 and they had an average of 145 segments (Quillin, 1998) yielding a length to radius ratio of 0.27. This ratio at a fiber angle of 54.7 deg in Eqn 43 implies that n=0.06, i.e. the helical fiber wraps each segment just 0.06 times. That a segment is wrapped only 0.06 times by a helical fiber at a 54.7 deg angle is a surprising and testable prediction of the assumed geometry. A longer structure such as a squid tentacle with a length to radius ratio of about 40 at that same angle (Kier and Smith, 1985) would be wrapped 9 times.

Assume a constant volume of body wall

The body wall deforms as a cylindrical hydrostat changes shape. To account for this deformation, we assume that the body wall is a constant volume and thin relative to the radius of the cylindrical body. The thickness of the body wall is therefore given by:
formula
(44)
with v being the volume of the body wall and subscript cvf indicating constant volume with cylindrically wrapped fibers. This body wall has a cross-sectional area:
formula
(45)
We assume that the volume of the body wall is a fraction κ of the body volume:
formula
(46)
Under these assumptions, the ratio of the body wall thickness (Eqn 44) to the radius (Eqn 46) is:
formula
(47)
which is a constant for all worm shapes. In our example calculations, we use κ=0.1, which means that the wall thickness is 0.05 times the radius.

The mechanics of linearly elastic fibers

Fiber extension depends on the material properties of the fibers. We assume a constant stiffness and hence a model of fibers that generate force in proportion to their extension beyond a resting length. Therefore, stress in a fiber with strain and Young's modulus E is:
formula
(48)
By substitution of fiber length (Eqns 38 and 39), the stress, with respect to the helix angle, is:
formula
(49)
For this expression, the components of stress along the hoop and longitudinal directions (Eqns 13 and 14) are:
formula
(50)
formula
(51)
We can relate the fluid pressure to the fiber stress. This is achieved through considering a balance of forces across a longitudinal plane through the worm, similar to the McKibben actuator (Eqn 15), but including the hoop stresses σ­m generated by circular muscles:
formula
(52)
The muscles here are assumed to produce a fixed level of stress, without consideration of the muscle's length–tension curves or force–velocity properties. Because muscles can control their level of activation, a muscle can control the stress produced, to some extent, independent of the point on the length–tension curve at which the muscle is operating. We are primarily interested in the way that a hoop fiber or muscle can carry some of the hoop stress, and we are interested in how that changes the angle at which forces are balanced. Therefore, we model only this one aspect of circular muscle action – the level of stress. This allowed us to generate graphs that are series of static balances with different fiber angles and output and input forces. The model we built can be subsequently modified to model muscle properties more specifically.
From the force balance across the longitudinal plane (Eqn 52), we incorporate our expressions for the hoop stress of fibers (Eqn 50) and the body dimensions (Eqns 40, 41 and 44):
formula
(53)
This relationship between pressure and body shape reveals some fundamental mechanical features. In the absence of circular muscle stress (σ­m­=0), the internal pressure is minimized when the body is at a circumference where θ=54.7 deg (Fig. 6D). Pressure is higher at fiber angles on either side of this minimum pressure angle. Thus, a worm that is extending or retracting will generate internal coelomic pressures consistent with experiments (Seymour, 1969; Quillin, 1998, 2000). Circular muscle contraction with concurrent elongation coincides with increases in coelomic pressure, whereas subsequent circular muscle relaxation and contraction of the longitudinal muscle cause a decrease in pressure as the angle returns to 54.7 deg.

The axial force with cross-helical fibers

The axial force is the input to the hydrostatic skeleton on retraction and the output on extension. This force could be due to a combination of longitudinal muscles, an external force, or stress within the fiber winding. We found an equation for the axial force by considering internal pressure and resistance generated by the fiber winding, which mirrors our derivation of output force by the McKibben actuator (Eqn 17), though with pressure potentially generated by muscular action:
formula
(54)
Substituting relationships for pressure (Eqn 53), area (Eqns 42 and 45) and longitudinal stress (Eqn 51) gives:
formula
(55)
Plotting the predictions of axial force illustrates some of the salient features of a fiber-wound hydrostat. The passive properties of this system may be considered for the case where the circular muscles generate no tension (violet curve in Fig. 6D,E). The body generates no axial force when helix angle is 54.7 deg. Despite the Hookean elasticity of the fibers, the helical geometry generates a non-linear relationship between circumference and force that elevates the restorative forces at the extremes. For example, body shortening expands the circumference and elevates fiber tension in a manner that generates a pushing force by the body, which will work to elongate the body back to its resting form like a non-linear spring.

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.

Mechanical advantage and displacement advantage on axial extension

Determining the mechanical advantage when the body extends requires a model for the input force. This force is generated by the circular muscles and is equal to the product of muscle stress and wall thickness and length of the cylindrical body (Fin,cvf=lcvftσ­m). Writing the length of the body as a function of the helix angle (Eqn 40) yields:
formula
(56)
The mechanical advantage is the ratio of axial force (Eqn 55) to input force (Eqn 56) for the extension by circular muscles:
formula
(57)
The DA on extension is a function of length and radius of the cylinder with respect to fiber angle (Eqns 40 and 41):
formula
(58)
The displacement advantage here is equivalent to the cylindrical hydrostat without fibers (Eqn 32). This is perhaps unsurprising given that a cylinder possesses only two variables, one that changes through muscular action and the other that is dictated by the equation for the volume of a cylinder (Eqns 40 and 41). Note that the DA here is not equal to the inverse of the MA (Eqn 57), which reflects the storage of elastic energy in the helical fibers.
The transmission efficiency is given (using Eqns 57 and 58) by:
formula
(59)

Mechanical and displacement advantage on axial retraction

Consider an output force in the radial direction, such as might be applied to the burrow wall of an earthworm. This is distinct from the input force on extension, which we assumed to be generated by circular muscles (Eqn 56). Thus, the mechanical advantage on retraction is the inverse of that for extension, times a geometrical factor (2π) that converts circumferential to radial forces:
formula
(60)
This relationship between the mechanical advantage in the two modes of operation for the worm can be seen graphically (Fig. 7D), where the mechanical advantage in retraction at 54.7 deg is 2π times higher than that of extension.
The displacement advantage for retraction may be derived similar to the fiberless model (Eqn 36), but incorporates how the length and radius of the cylinder change with helix angle (Eqns 40 and 41):
formula
(61)
As for extension, the displacement advantage here is the same as that for the cylindrical hydrostat without fibers (Eqn 36). As for the fiberless model, the displacement advantage for retraction is proportional to the inverse of that for extension []. Given this relationship and Eqn 60, the transmission efficiency for retraction is:
formula
(62)
Therefore, the transmission efficiency for retraction is simply the inverse that for extension (Eqn 59).

Patterns of mechanical advantage and transmission efficiency in cylindrical hydrostats with cross-helical fibers

Consider a given cylinder (constant n, E) with a specified σm. 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.

Mechanical advantage at 54.7 deg

The mechanical advantage at 54.7 deg is much greater on retraction than extension in the example plotted in Fig. 7, but in general this will not be the case. Specifically, using Eqns 43, 57 and 60, we can relate mechanical advantage to the number of windings n and to the length to radius ratio:
formula
(63)
formula
(64)
formula
(65)
where l54 and r54 are the length and radius of a constant-volume, cylindrical, fiber-wound hydrostat when fiber angle is 54.7 deg. Note that at 54.7 deg, the mechanical advantage formulas for constant-volume fiber-wound hydrostats are identical to those for constant-volume fiberless hydrostats (compare with Fig. 5 and Eqns 35, 37). Furthermore, the ratio of mechanical advantage in retraction compared with extension at 54.7 deg (i.e. at a given shape) depends on n2 (Eqn 65). This means that mechanical advantage in retraction will always be larger than in extension when the cylinder at 54.7 deg has n>1/(2√π)≈0.28. This number of fiber wrappings per length occurs at a length to radius ratio, solving from Eqn 43, of √(π/2)≈1.25. Longer, thinner cylinders at 54.7 deg will have higher mechanical advantage in retraction (i.e. generate higher radial output forces relative to axial input forces) and shorter, wider cylinders will have a higher mechanical advantage in extension (i.e. generate higher axial output forces than input circular forces). The example plotted in Fig. 7 has n=1 and thus has a length to radius ratio of (√2)π≈4.4 (Eqn 43). It will be interesting to compare length to radius ratios of biological cylinders operating close to 54.7 deg to see whether the length to radius ratio confers a higher mechanical advantage in retraction or extension. It will also be interesting to compare the predictions about mechanical advantage from the theory with and without helical fibers, a comparison we will undertake in the Discussion.

Comparing mechanical advantage in a fiber-wound constant-volume hydrostat with a fiberless constant-volume hydrostat

To compare the predictions of these two hydrostat models, we calculate mechanical advantage in extension and retraction (Eqns 43, 57 and 60), but with the assumption that the fibers have no stiffness. The lack of stiffness makes it impossible for the fibers to affect the mechanics by storing energy. The mechanical advantage under zero stiffness (E=0) is:
formula
(66)
formula
(67)
where lE=0 and rE=0 are the length and radius of a constant-volume fiber-wound cylindrical hydrostat with fibers with zero or low stiffness. We can see that when the fibers are not stiff, the mechanical advantage for fiber-wound constant-volume hydrostats is the same as that predicted for fiberless constant-volume hydrostats in Fig. 5 (see also Eqns 35 and 37). To further estimate the effects of muscle stress and fiber stiffness, one can plot an overlay of predictions from fiber-wound and fiberless cases. First, plot MA,cvf and MAℝ,cvf from Eqns 57 and 60 against the length to radius ratio from Eqn 43, all of which are parametric functions of θ. Then overlay a plot of MA,cv and MAℝ,cv from Eqns 35 and 37 against the length to radius ratio. On such plots, one can vary fiber stiffness and fiber stress to discover that the higher the muscle stress relative to the fiber stiffness, the closer the predictions of mechanical advantage in the fiber-wound and fiberless cases. One can also see that higher fiber stiffness generates lower mechanical advantage at a given length to diameter ratio. We leave this plot as an exercise for the reader.

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.

Variable gearing

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.

Insight can also be gained into earthworm mechanical behavior by comparing mechanical advantage in extension and retraction for earthworm segments. From formulas for fiberless constant-volume hydrostats in Fig. 5 (Eqns 35 and 37):
formula
(68)
By plugging in 0.27, calculated from Quillin (1998), for l/r of an earthworm segment in Eqn 68, we infer that the mechanical advantage in retraction in earthworms is just 0.046 times the mechanical advantage in extension. This direct comparison between mechanical advantage in retraction versus extension has not previously been made (Quillin, 1998; Kurth and Kier, 2014, 2015). We used a calculation for constant-volume fiberless hydrostats to make this inference; however, these earthworms are fiber wound. However, at 54.7 deg, we showed that the ratio of mechanical advantage predicted for fiber-wound and fiberless constant-volume hydrostats is the same (compare with Eqn 65). This equivalence at 54.7 deg should not be too surprising because we know that at 54.7 deg the fiber-wound cylinder is at equilibrium and the fibers are not stretched and not storing energy. Thus, for angles close to 54.7 deg we can infer that earthworm segment dimensions generate more mechanical advantage in extension than in retraction. One intriguing possibility is that earthworms can vary the mechanical advantage on extension and retraction by cooperating among segments. To the degree that segments may behave as longer coordinated groups, they might be able to adjust the effective length to gain mechanical advantage in extension, but this proposition requires experimental testing.

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.

The relationship between mechanical advantage and displacement advantage

Energy relates mechanical and displacement advantage. If output work (loutFout) generated by a skeleton is equal to its input work (linFin), 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.

Geometric modeling of hydrostatic skeletons

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.

The mechanics of helical fiber winding

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.

Biological relevance and testable predictions

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.

Author contributions

Conceptualization: M.J.M., O.E., A.S.J.; Methodology: O.E., K.-I.E., A.S.J., T.P., S.H., E.K., M.J.M.; Writing - original draft: M.J.M., O.E., A.S.J.; Writing review & editing: O.E., K.-I.E., A.S.J., T.P., S.H., E.K., M.J.M.; Visualization: M.J.M., O.E., A.S.J.; Supervision: M.J.M.; Project administration: M.J.M.; Funding acquisition: M.J.M.

Funding

This project was supported by grants from the National Science Foundation (IOS-2034043, IOS-2326484) and the Office of Naval Research (N00014-17-1-2062 and N00014-19-1-2035). Open access funding provided by University of California, Irvine. Deposited in PMC for immediate release.

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Competing interests

The authors declare no competing or financial interests.

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