TY - JOUR
AU - Azizi,, Emanuel
AB - Abstract Systems powered by elastic recoil need a latch to prevent motion while a spring is loaded but allow motion during spring recoil. Some jumping animals that rely on elastic recoil use the increasing mechanical advantage of limb extensor muscles to accomplish latching. We examined the ways in which limb morphology affects latching and the resulting performance of an elastic-recoil mechanism. Additionally, because increasing mechanical advantage is a consequence of limb extension that may be found in many systems, we examined the mechanical consequences for muscle in the absence of elastic elements. By simulating muscle contractions against a simplified model of an extending limb, we found that increasing mechanical advantage can limit the work done by muscle by accelerating muscle shortening during limb extension. The inclusion of a series elastic element dramatically improves mechanical output by allowing for additional muscle work that is stored and released from the spring. This suggests that elastic recoil may be beneficial for more animals than expected when assuming peak isotonic power output from muscle during jumping. The mechanical output of elastic recoil depends on limb morphology; long limbs moving small loads maximize total work, but it is done at a low power, whereas shorter limbs moving larger loads do less work at a higher power. This work-power trade-off of limb morphology is true with or without an elastic element. Systems with relatively short limbs may have performance that is robust to variable conditions such as body mass or muscle activation, while long-limbed systems risk complete failure with relatively minor perturbations. Finally, a changing mechanical advantage latch allows for muscle work to be done simultaneously with spring recoil, changing the predictions for spring mechanical properties. Overall, the design constraints revealed by considering the mechanics of this particular latch will inform our understanding of the evolution of elastic-recoil mechanisms and our attempts to engineer similar systems. Introduction Many biological and engineered movements are actuated by recoiling elastic structures (Peplowski and Marsh 1997; Aerts 1998; Armour et al. 2007; Kovac et al. 2008; Buksh et al. 2010; Haldane et al. 2016). These systems are characterized by four components: a motor that provides the energy of movement; a spring that stores and releases energy; a latch that controls the timing and rate of spring recoil; and a projectile that is ultimately accelerated (Ilton et al. 2018). Each component possesses properties that influence the dynamics of the system, and components must be tuned to one another in order to achieve a particular mechanical outcome (Ilton et al. 2018). For biological systems, we may expect selection to shape the morphology and physiology of structures filling each role to result in a tuned, integrated mechanism that achieves a performance goal in some ecologically relevant behavior (Bock 1980; Arnold 1983). Latches possess several properties that can influence performance of the system and can vary widely in their structure and method of action (Sakes et al. 2016); therefore, the ways in which they impact whole-system performance must also vary. Latches have two important roles in an elastic mechanism: first, they must prevent motion while the motor builds forces and stores energy in the spring, and then they must allow motion so that spring recoil can accelerate the projectile. For example, “anatomical latches” are physical structures that oppose motion of the projectile or recoil of the spring until they are removed at a critical time. The rate at which an anatomical latch is removed can drastically alter projectile velocity by gating energy release from the recoiling spring (Ilton et al. 2018). These types of latches, where the prevention and release of spring motion are clearly observable, are common for invertebrate systems (Gronenberg et al. 1993), but the dual roles of latching can be accomplished by other means in some systems. Frogs that jump using elastic recoil are hypothesized to use a changing mechanical advantage latching mechanism (Roberts and Marsh 2003; Astley and Roberts 2012,, 2014; see section “Methods”) and this mechanism could apply to many vertebrate muscle–tendon units used for jumping. Poor mechanical advantage of the limb extensor muscles initially prevents motion and allows for energy to be stored in series elastic elements, whereas increasing mechanical advantage during limb extension allows for release of stored energy into motion. The timing of loading and unloading using increasing mechanical advantage arises from the properties and arrangement of the motor, spring, and projectile, including linked segments. Therefore, tuned performance of the whole system depends on the interactions between the components in a compounding fashion. We can expect complex interactions where motor, spring, and projectile properties influence the process of both loading and unloading, which then in turn impacts the efficacy of projectile acceleration by the motor and spring. Dynamic changes in mechanical advantage are present in some form in any animal that jumps by extending its limbs, though latching behavior may depend on morphological variation (Astley and Roberts 2014). Despite the wide variation in morphology and physiology of locomotor systems in these animals, it is not yet known how properties of the motor (muscle), spring (elastic structures), and projectile (body mass and limb morphology) determine mechanical advantage latching or how this determines performance of the integrated system. Additionally, although not all animals may store elastic energy during jumping, it is unknown how changing mechanical advantage influences projectile motion under direct muscle actuation. Locomotion via extending limbs is common across the animal kingdom, underpinning many major modes of movement; therefore, changing mechanical advantage may be an important determinant of musculoskeletal performance regardless of the role of elastic structures. Here we explore the tuning of the motor, spring, and projectile, emergent latching behavior, and resulting performance in systems with mechanical advantage latching. We model muscle contraction in a simple simulated limb joint with varying spring and projectile properties. We start by investigating the impact of projectile morphology on latching behavior and performance without an elastic spring, and then we explore the tuning of spring stiffness and projectile morphology in the same system. Methods We constructed a simplified version of an extending limb in Simulink through a bespoke MATLAB (vR2017, The MathWorks, Inc., Natick, MA) interface. The model consisted of a single frictionless joint between two rigid, massless segments forming a first-class lever (Fig. 1). A muscle positioned parallel to the upper segment applied input force, Fm, at the end of the in-lever, Lin. The output force was modeled as applied only in the vertical direction reflecting the role of a foot in mediating force application to the substrate in a living system. Force applied to the rigid substrate resulted in an equal and opposite ground reaction force (GRF). Fig. 1 Open in new tabDownload slide Schematic of the modeled limb joint used in simulated muscle contractions. Two rigid massless segments articulate to form a lever system with Lin, and Lout. A muscle positioned parallel with the upper segment applies force, Fm, resulting in an output force and equal, opposite, GRF. EMA of the muscle is defined as the ratio of the in-moment arm, r, to the out-moment arm, R. The GRF accelerates a gravitational load at the end of the upper segment. The length of the muscle was modeled independently of the length of the upper segment and GRF was modeled to always result in a purely vertical acceleration of the load mass to simplify calculations. Fig. 1 Open in new tabDownload slide Schematic of the modeled limb joint used in simulated muscle contractions. Two rigid massless segments articulate to form a lever system with Lin, and Lout. A muscle positioned parallel with the upper segment applies force, Fm, resulting in an output force and equal, opposite, GRF. EMA of the muscle is defined as the ratio of the in-moment arm, r, to the out-moment arm, R. The GRF accelerates a gravitational load at the end of the upper segment. The length of the muscle was modeled independently of the length of the upper segment and GRF was modeled to always result in a purely vertical acceleration of the load mass to simplify calculations. Effective mechanical advantage (EMA) describes the ratio of GRF to Fm, and can be expressed as the ratio between the perpendicular distance from Fm to the center of joint rotation (r) and the perpendicular distance from GRF to the center of joint rotation (R) (Fig. 1). The Fm acting through EMA resulted in a GRF that accelerated a gravitational load mass positioned at the end of the upper segment. The load mass was not allowed to accelerate with a net negative force to prevent flexion of the joint angle (θ) past an initial 10°. A net positive force accelerated the mass upward, extending the joint. As the simulation proceeded through time at 1000 Hz, acceleration of the mass and joint extension led to increasing EMA because both r and R are dependent on θ. Simulations continued until Fm dropped to zero representing takeoff from the ground or θ exceeded 180° representing full joint extension. The muscle was modeled using the contractile properties of a 1.5–2.0 g bullfrog plantaris longus muscle (Table 1). These included Hill-type equations for force–length and force–velocity properties of the contractile element (CE), as well as time-dependent activation. A series elastic element (SEE) was included in the muscle as a Hookean spring with a stiffness, k (Fig. 1). Simulated tetanic contractions were first generated for models with no SEE (i.e., k > 1010 Nm−1) with load masses from 0.0004 to 0.02 kg and out-lever, Lout from 0.004 to 0.2 m. Additional simulations were then run for a subset of load masses and Lout while k varied from 400 to 20,000 Nm−1, a biologically relevant range based on measured values for frog tendon (Sawicki et al. 2015; Rosario et al. 2016). Table 1 Muscle properties used for simulations L0 (m) . Vmax (ms−1) . P0 (N) . Time to 100% activation (s) . In-lever (m) . Starting length (L0) . 0.011 0.124 15 0.1 0.002 1.1 L0 (m) . Vmax (ms−1) . P0 (N) . Time to 100% activation (s) . In-lever (m) . Starting length (L0) . 0.011 0.124 15 0.1 0.002 1.1 Open in new tab Table 1 Muscle properties used for simulations L0 (m) . Vmax (ms−1) . P0 (N) . Time to 100% activation (s) . In-lever (m) . Starting length (L0) . 0.011 0.124 15 0.1 0.002 1.1 L0 (m) . Vmax (ms−1) . P0 (N) . Time to 100% activation (s) . In-lever (m) . Starting length (L0) . 0.011 0.124 15 0.1 0.002 1.1 Open in new tab Results Dynamics with and without elastic elements At the start of the simulations, EMA was poor due to the small θ and long Lout. Motion did not occur until the force generated by the CE, multiplied with EMA, was greater than the force of gravity acting on the load mass. Because it takes time for CE to build force, the system remained motionless until this force threshold was reached. As the force applied by the CE through EMA exceeded the force of gravity, motion began slowly at first. The joint extended some amount because of the upward displacement of the load mass experiencing a net positive force. This joint extension reduced R while increasing r, leading to an increasing EMA and making further application of Fm more effective at accelerating the load mass. Thus, the system underwent positive feedback where extension of the joint led to greater projectile displacement and greater joint extension and the system accelerated in a characteristic fashion (Fig. 2A). Fig. 2 Open in new tabDownload slide Example of force–length traces for a simulated muscle contraction against the modeled limb joint (A) without a series elastic element and (B) with a series elastic element. Data were simulated at 1000 Hz, thus the density of data point represents the velocity of muscle contraction. The solid line is the isometric force–length relationship of the modeled muscle. Schematics indicate the action of the limb joint at sample time points throughout the muscle contraction. Fig. 2 Open in new tabDownload slide Example of force–length traces for a simulated muscle contraction against the modeled limb joint (A) without a series elastic element and (B) with a series elastic element. Data were simulated at 1000 Hz, thus the density of data point represents the velocity of muscle contraction. The solid line is the isometric force–length relationship of the modeled muscle. Schematics indicate the action of the limb joint at sample time points throughout the muscle contraction. The entire event was characterized by two phases: a stationary phase in which force developed but there was no movement about the joint, and an acceleratory phase in which the joint extended under muscle force. When no SEEs were present, the CE developed force in the stationary phase, but no work was done because there was no shortening of the CE (Fig. 2A). During the acceleratory phase, the shortening velocity of the CE increased thereby rapidly reducing Fm because of the force–velocity trade-off of muscle. When elastic elements were present, the muscle was able to shorten against the lengthening SEEs during the stationary phase (Fig. 2B). Thus work done by the CE was stored as elastic potential energy (PE) in the stretched SEE. During the acceleratory phase, the extending joint led to increased EMA, lowering the force threshold necessary to overcome gravity and continue acceleration of the projectile. The SEE recoiled as the force threshold dropped, shortening so Ltendon-Lslack=Fm/k, and releasing the stored elastic energy. During this time, the CE continued to shorten at a force equal to that of the SEE and a velocity determined by the force–velocity relationship, doing additional work to accelerate the projectile. Effects of limb morphology Increasing the load mass or Lout both increased the force threshold that the CE must reach to transition from the stationary to the acceleratory phase (Fig. 3A). The total work done by the CE was highest in cases where the force threshold was just below the maximum force the CE can generate based on the force–length properties of muscle (Fig. 3B). For every load mass there was an Lout that resulted in a force threshold near but just below the CE maximum, all else remaining the same (Fig. 3B). Therefore, for all investigations of the effects of limb morphology, Lout was determined as a function of load mass by solving the following equations: F = k(1.1L0-Lm) (1) F=P0e-|((Lm/L0(-2.89-1)-0.75)2.08| (2) to determine the force (F) at which the muscle contracting to stretch a SEE of stiffness, k, from a starting length of 1.1 L0 (Azizi and Roberts 2010; Equation (1)) would intersect the force–length relationship of the muscle (Equation (2)). The EMA required to produce a threshold of 95% F was determined and the Lout that produced that EMA given a starting angle of 10° was found. Fig. 3 Open in new tabDownload slide (A) Force–length traces from simulated muscle contractions against the modeled limb joint with no series elastic elements. For each trace, all parameters were the same except for load mass. Trace 3 represents biologically reasonable values of out-lever length and projectile mass considering the properties of the modeled muscle. Data were simulated at 1000 Hz, thus the density of data point represents the velocity of muscle contraction. The solid line is the isometric force–length relationship of the modeled muscle. Because of acceleration due to gravity, there is some minimum muscle force threshold that must be reached in order for motion to occur and this threshold is determined by the morphology of the limb joint. For a given set of model parameters, including out-lever length, there is a load mass that maximizes the work done by the muscle (area under curve) without reaching the limits of muscle force production at the operating length. (B) Energy (work) done by the muscle contraction as a function of out-lever length and load mass. For any out-lever length, there is some load mass that maximizes the work done by the muscle by raising the force threshold for motion to just below the peak force of muscle at that operating length. When that load mass is exceeded, the force threshold required for motion is greater than the peak force of muscle at that length, no motion occurs, and no work is done. Thus, for a given elastic element stiffness and muscle force–length relationship, there is an out-lever length that maximizes work done for each load mass (dashed line). Fig. 3 Open in new tabDownload slide (A) Force–length traces from simulated muscle contractions against the modeled limb joint with no series elastic elements. For each trace, all parameters were the same except for load mass. Trace 3 represents biologically reasonable values of out-lever length and projectile mass considering the properties of the modeled muscle. Data were simulated at 1000 Hz, thus the density of data point represents the velocity of muscle contraction. The solid line is the isometric force–length relationship of the modeled muscle. Because of acceleration due to gravity, there is some minimum muscle force threshold that must be reached in order for motion to occur and this threshold is determined by the morphology of the limb joint. For a given set of model parameters, including out-lever length, there is a load mass that maximizes the work done by the muscle (area under curve) without reaching the limits of muscle force production at the operating length. (B) Energy (work) done by the muscle contraction as a function of out-lever length and load mass. For any out-lever length, there is some load mass that maximizes the work done by the muscle by raising the force threshold for motion to just below the peak force of muscle at that operating length. When that load mass is exceeded, the force threshold required for motion is greater than the peak force of muscle at that length, no motion occurs, and no work is done. Thus, for a given elastic element stiffness and muscle force–length relationship, there is an out-lever length that maximizes work done for each load mass (dashed line). The same force threshold could be reached by a small load mass actuated through a long Lout, or a large load mass actuated through a short Lout. A shorter Lout for a given load mass resulted in less work done during contraction, but importantly, an Lout that was too long for a given load mass prevented any motion from occurring because the force threshold was above the maximum force the CE could generate at that length (Fig. 3). For a given CE starting length, Lin, and contractile properties, a small load mass actuated through a long Lout resulted in greater muscle work than a large mass actuated through a short Lout (Fig. 4). A long Lout kept EMA poorer throughout the contraction, leading to a lower acceleration. Thus, the CE operated at a slower velocity and higher force for a greater proportion of the motion compared with the rapid acceleration seen in large-mass/short Lout contractions (Fig. 4). Although the work done by the CE was greater in a low-mass/long Lout contraction, the duration of this motion was longer, resulting in a lower average power (Fig. 5). This tradeoff between work and power was true regardless of the starting conditions for the contraction or the present/absence of a SEE (Fig. 5). Fig. 4 Open in new tabDownload slide Example of force–length trajectories from simulated muscle contractions against the modeled limb joint with no series elastic elements. Data were simulated at 1000 Hz, thus the density of data point represents the velocity of muscle contraction. The solid line is the isometric force–length relationship of the modeled muscle. INSET: the set of out-lever lengths and load masses that maximize the energy output of the muscle (Fig. 3B). Force–length traces represent the extremes of the load mass/out-lever length relationship. Note that although the work done (area under curve) is greater with a long out-lever and low load mass (2), this contraction proceeds more slowly than the contraction with a short out-lever and high load mass (1). Fig. 4 Open in new tabDownload slide Example of force–length trajectories from simulated muscle contractions against the modeled limb joint with no series elastic elements. Data were simulated at 1000 Hz, thus the density of data point represents the velocity of muscle contraction. The solid line is the isometric force–length relationship of the modeled muscle. INSET: the set of out-lever lengths and load masses that maximize the energy output of the muscle (Fig. 3B). Force–length traces represent the extremes of the load mass/out-lever length relationship. Note that although the work done (area under curve) is greater with a long out-lever and low load mass (2), this contraction proceeds more slowly than the contraction with a short out-lever and high load mass (1). Fig. 5 Open in new tabDownload slide The (A) energy, (B) movement duration (excluding loading/force development), and (C) average movement power for simulations with a spring (solid) and without a spring (dashed) as a function of load mass. For each load mass, data were simulated with the corresponding out-lever length that maximized the work done by the muscle, that is, low load mass with long out-lever and high load mass with short out-lever (Fig. 3). Fig. 5 Open in new tabDownload slide The (A) energy, (B) movement duration (excluding loading/force development), and (C) average movement power for simulations with a spring (solid) and without a spring (dashed) as a function of load mass. For each load mass, data were simulated with the corresponding out-lever length that maximized the work done by the muscle, that is, low load mass with long out-lever and high load mass with short out-lever (Fig. 3). Effects of SEE stiffness When contracting against a SEE, the CE shortened during the stationary phase changing the potential peak force it could reach based on the force–length properties of muscle. The stiffness of the SEE determined where the CE intersected its force–length curve and thus the peak force that could be exerted on the SEE before shortening stopped. The load mass and Lout combinations that set a force threshold just below the peak force of the muscle depended on the stiffness of the SEE (Fig. 6). For a given Lout, a stiffer spring could move a larger load mass and do more work compared with a spring that optimized energy storage. A SEE with k = 5000 Nm−1 stored the most elastic energy given this set of muscle properties, but an SEE with k = 8000 Nm−1 gave the most total work because a higher force threshold was reached allowing the CE to do more additional work during the acceleratory phase. Fig. 6 Open in new tabDownload slide (A) The force–length trajectory for k = 5000 Nm−1, load mass = 0.008 kg, out-lever = 0.0491 m, (B) k = 8000 Nm−1, load mass = 0.010 kg, out-lever= 0.0491 m, and (C) k = 8000 Nm−1, load mass = 0.008 kg, out-lever = 0.0491 m. Shaded regions represent energy stored in series elastic elements that is later released into motion of the load mass. Hashed regions represent work done by muscle contraction directly simultaneous with recoil of the series elastic elements. For this muscle, spring stiffness of 5000 Nm−1 (A) maximizes the stored and released elastic energy, but a spring stiffness of 8000 Nm−1 (B) maximizes the total work because the higher forces reached by the muscle allow for more direct muscle work. Because the muscle can reach higher forces, the muscle with a stiffer spring can do more work while moving a heavier load mass (B vs. C). Fig. 6 Open in new tabDownload slide (A) The force–length trajectory for k = 5000 Nm−1, load mass = 0.008 kg, out-lever = 0.0491 m, (B) k = 8000 Nm−1, load mass = 0.010 kg, out-lever= 0.0491 m, and (C) k = 8000 Nm−1, load mass = 0.008 kg, out-lever = 0.0491 m. Shaded regions represent energy stored in series elastic elements that is later released into motion of the load mass. Hashed regions represent work done by muscle contraction directly simultaneous with recoil of the series elastic elements. For this muscle, spring stiffness of 5000 Nm−1 (A) maximizes the stored and released elastic energy, but a spring stiffness of 8000 Nm−1 (B) maximizes the total work because the higher forces reached by the muscle allow for more direct muscle work. Because the muscle can reach higher forces, the muscle with a stiffer spring can do more work while moving a heavier load mass (B vs. C). Although total energy was greatest in simulated contractions against stiffer springs with low load mass and long Lout, these extreme configurations are unlikely to be recreated in physical systems (see Discussion). For larger load masses and shorter Lout, k ranging from 4000 to 10,000 Nm−1 gave the greatest total energy (Fig. 7A). The energy at time of takeoff (either when Fm = 0 or θ  >  180°) was composed of the change in gravitational PE resulting from extension of the joint and the kinetic energy (KE) of the load mass. At values of k < 6000 Nm−1, Fm reached zero before the joint had fully extended, reducing the change in PE before takeoff (Fig. 7B). For k > 12,000 Nm−1, the limb joint fully extended at or before the time that Fm reached zero. Because of the inverse relationship between load mass and Lout, full extension of the joint resulted in the same change in PE for all load masses, that is, smaller masses moved a longer distance and larger masses moved a shorter distance. For intermediate values, the amount of joint extension and change in PE depended on the load mass being actuated; with larger load masses, Fm reached zero and takeoff was achieved before the joint completely extended (Fig. 7B). Patterns of KE mirror those of total energy with the greatest values for larger load masses and shorter Lout, and k ranging from 4000 to 10,000 Nm−1 (Fig. 7C). Fig. 7. Open in new tabDownload slide Total energy of motion (stored elastic energy and direct muscle work) (A), gravitational potential energy of the load mass at takeoff (B), and KE of the load mass at takeoff (C) as a function of spring stiffness and load mass. For each load mass, data were simulated with the corresponding out-lever length that maximized the work done by the muscle, i.e. small load mass with long out-lever and large load mass with short out-lever (Fig. 3; note that this relationship is non-linear, thus values of out-lever length are not plotted on the x-axis). Although a spring stiffness of 5000 Nm−1 maximizes the stored elastic energy for this muscle, total energy of motion is always greater when using a stiffer spring at any load mass. Fig. 7. Open in new tabDownload slide Total energy of motion (stored elastic energy and direct muscle work) (A), gravitational potential energy of the load mass at takeoff (B), and KE of the load mass at takeoff (C) as a function of spring stiffness and load mass. For each load mass, data were simulated with the corresponding out-lever length that maximized the work done by the muscle, i.e. small load mass with long out-lever and large load mass with short out-lever (Fig. 3; note that this relationship is non-linear, thus values of out-lever length are not plotted on the x-axis). Although a spring stiffness of 5000 Nm−1 maximizes the stored elastic energy for this muscle, total energy of motion is always greater when using a stiffer spring at any load mass. Discussion Effects of changing mechanical advantage A mechanical advantage latch forces muscle to contract in a way that is suboptimal for meeting the challenges of jumping. If jump performance is determined by average power—releasing the most energy in the shortest amount of time—then the best muscle contraction would shorten at peak power for the duration of the jump. Increasing mechanical advantage accelerates muscle contraction, keeping it from spending much time at or near peak power. A decreasing mechanical advantage would allow the muscle to contract with a high power output for the duration of motion (Carrier et al. 1994), but this is impossible given the pattern of limb joint extension during jumping. Performance in a system with increasing mechanical advantage is improved by including a series elastic element. Energy stored when mechanical advantage is low and the system is stationary is recovered when the elastic element recoils during the acceleratory phase (Fig. 2). Depending on the stiffness of the elastic element, the majority of muscle work can move through the elastic element. Because the force of the elastic element is largely independent of recoil velocity, this energy can be released at a higher power than the contractile element could generate. Therefore, the inclusion of elastic elements with sufficient stiffness improves both the work and power output of muscle working against an increasing mechanical advantage (Fig. 5). Elastic recoil may be more common than previously observed because of the widespread nature of morphology that results in increasing mechanical advantage during joint extension. Elastic recoil is often identified in a system when the maximum power of a movement exceeds the upper limits of muscle under isotonic contraction (Evans 1973; Patek et al. 2004; Lappin et al. 2006; Deban et al. 2007; Van Wassenbergh et al. 2008; Burrows 2013). However, a mechanical advantage latch limits the power of direct muscle contraction so that it may only transiently reach peak isotonic power, if at all. While a comparison of movement power to peak isotonic muscle power may suggest that a motion can be accomplished by muscle alone, comparing movement power to what the muscle can achieve in a realistic contraction, for example, with increasing mechanical advantage, could reveal the presence of elastic actuation. Although elastic recoil is often associated with jumping in small animals (Evans 1973; Heitler 1977; Burrows 2009,, 2013; Sutton and Burrows 2011) that must accelerate faster than larger animals to compensate for briefer contact with the ground (Alexander 1988; Gronenberg 1996), the effects of increasing mechanical advantage in extending limbs may also make elastic recoil beneficial in larger animals. Elastic actuation is important for small animals when the scaling of morphology imposes extreme power demands during jumping that cannot be met by muscle contraction (Bennet-Clark and Lucey 1967; Roberts et al. 2011). These arguments identify a threshold size below which direct muscle actuation would be insufficient based on the maximum isotonic power output. Sutton et al. (2019) demonstrate that because only a portion of potential muscle work can be stored in a linear spring, direct muscle actuation produces higher performance jumps above a certain body size. Under this reasoning, muscle is not only capable of meeting the power demands of jumping at larger body sizes, but also results in greater performance than could be achieved using elastic recoil. Considering that increasing mechanical advantage prevents muscle from contracting with high average power, the body threshold size above which muscle actuation is more effective than elastic recoil may be larger than previously modeled (Sutton et al. 2019). Tuning the mechanical advantage latch Performance is limited when no elastic element is present, but limb morphology can be altered to increase the work being done. Specifically, a low load mass actuated through relatively long limbs will slow down muscle contraction, allowing for shortening at high forces and the greatest amount of work. However, this slower shortening leads to a longer duration of movement and lower average power. Although the inclusion of elastic elements improves both the work and power output well beyond direct muscle actuation, the tradeoff between work and power based on morphology remains (Fig. 5). It is important to note that although, the lower limits of load mass and upper limits of Lout used in these simulations give the greatest total work, they are unlikely to occur together in biological systems. The smallest load mass simulated was 0.0004 kg working to move a lever system with Lout of 0.3316 m. Even if 100% of the body mass were devoted to muscle, it is unlikely that a muscle of that size could have the contractile properties used in this model. Additionally, it is impossible to lengthen the limb without adding mass. Nonetheless, these simulations demonstrate the performance that can result from minimizing the mass of the body relative to muscle force production and lengthening the limb while minimizing concomitant increases in mass. It may be better to operate with a large load mass and relatively shorter limbs because of the relationship between limb length and the mass that gives the highest forces (Fig. 3B). Although high forces lead to the most work being done, if the force threshold of a mechanical advantage latch exceeds muscle limits at that length then no movement occurs. Limb length is relatively fixed within an individual, but body mass, muscle mass, and muscle activation can be highly variable over short timescales. When operating with relatively long limbs, any small change to mass or muscle activation can easily shift the force threshold for motion beyond the limits of muscle. In contrast, with relatively short limbs a system can accommodate changes in mass or muscle activation with a comparably small change in performance (Fig. 3B). Longer limbs can be beneficial for jumping, but limbs that are too long can severely limit performance. Long limbs have been long associated with jumping in frogs, primarily with the explanation that they increase the amount of time that forces can be exerted on the ground during takeoff (Marsh 1994). While this is true, these simulations show that long limbs also decrease the average power with which muscles are doing work during takeoff. If jumping demands balancing total energy with total duration, extremely long limbs could be detrimental. Additionally, long limbs are only useful if they are not so long that they create a force threshold that is too high for the muscle to reach under anything below optimal mass and activation conditions. A mechanical advantage latch changes the predictions for muscle–spring tuning compared with anatomical latches. Although a spring stiffness of 5000 Nm−1 is predicted to maximize stored elastic energy, a stiffer spring (8000 Nm−1) actually maximizes the total energy of the motion (Fig. 7). Unlike systems with an anatomical latch, mechanical advantage latched systems always have a combination of muscle work and elastic energy release during the acceleratory phase. The stiffer spring stores less elastic energy, but by reaching higher forces the muscle can do greater work while it is shortening during elastic recoil. The exact spring stiffness that is best for elastic energy storage may depend on the starting length of the muscle, but a relatively stiffer spring should always give the greatest total energy. Finally, the stiffer spring resulting in a higher force threshold allows for a greater mass to be moved for a given limb length (Fig. 7). Thus, a relatively stiffer spring may be better able to accommodate changes in mass or muscle activation levels while maintaining high performance. Limitations of the model Although the results presented here demonstrate the influence of morphology on the dynamics of mechanical advantage latching, the model is necessarily simple and omits many features that are present in complex living things. Predicted performance from simulations falls short of jumps observed from animals with similar dimensions and muscle contractile properties. If limb extensor muscle mass is ∼16% of body mass, then an appropriate load mass for the 1.5–2.0 g muscle in the model would be 0.0093–0.0125 kg. The associated Lout for this load mass that sets a force threshold near peak muscle force is between 0.0385 and 0.0513 m for k = 8000 Nm−1; the length of the lower hind limb segment of a bullfrog is roughly 0.030 m. Simulations with those parameters give a total muscle-mass-specific energy of 8.82–12.21 Jkg−1, which would translate to takeoff velocities of 1.29–1.53 ms−1, somewhat lower than reported values for Rana catesbeiana of 2.53 ms−1 (Marsh 1994). Some organisms possess characteristics that improve performance of the elastic recoil mechanism beyond what is seen in our simulations. For example, in a system with multiple limb joints, the torque generated by proximal joints opposes motion in the more distal joint; extensor muscles must generate forces to resist joint flexion resulting from more proximal muscle action (Astley and Roberts 2014). Thus, the torques generated by proximal joints can prevent motion and extend the loading phase, increasing stored elastic energy beyond that which would be stored against a relatively small gravitational load. Alternatively, many animals jump by rolling the point of contact with the ground distally while extending joints in the foot and digits (Carrier et al. 1994; Le Pellec and Maton 2002). This would effectively increase Lout throughout motion, keeping EMA poorer, allowing for both a longer loading phase and slower muscle contraction during unloading. Regardless of the mechanism, living animals likely possess latches upon latches, each controlling the storage and release of energy with complex temporal patterns determined by interactions of multiple morphological and physiological properties. Some limbs and body arrangements may result in different patterns of changing mechanical advantage during joint extension. Specialized jumpers, like frogs and kangaroos, have a posteriorly positioned center of mass (COM) and the GRF directed through the COM aligns with the limb so that joint extension results in increasing mechanical advantage at all joints. However, other jumping animals, particularly quadrupeds, have a more anteriorly positioned COM and the orientation of the GRF may only result in increasing mechanical advantage at the most distant joints (Carrier et al. 1998). In movements other than vertical jumping, distal translation and changes in orientation of the GRF can lead to decreasing EMA (Carrier et al. 1994, 1998). Although the current model does not represent these different patterns of changing mechanical advantage, it is clear that the lever mechanics resulting from jointed limbs is an important factor for understanding how muscle contraction results in useful work during movement. Conclusions By modeling mechanical advantage latching, we have revealed critical design constraints of elastic-recoil mechanisms, reflecting the importance of considering latch properties when studying or designing these systems. The unique way in which mechanical advantage latching arises from the interaction of the motor and projectile determines the tuning relationships of these component properties that result in a high-performance mechanical output. This is especially important in biological systems where increasing mechanical advantage may be present regardless of the use of elastic structures. Considering that increasing mechanical advantage may be a widespread consequence of limb morphology, these results have expanded the range of potential cases where muscle actuation is not sufficient to achieve observed performance. Our results predict that upon further examination, we may find elastic actuation contributing to movement in systems with otherwise mundane performance. From the symposium “Playing with Power: Mechanisms of Energy Flow in Organismal Movement” presented at the annual meeting of the Society for Integrative and Comparative Biology, January 3–7, 2019 at Tampa, Florida. 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TI - Modeling the Determinants of Mechanical Advantage During Jumping: Consequences for Spring- and Muscle-Driven Movement
JF - Integrative and Comparative Biology
DO - 10.1093/icb/icz139
DA - 2019-12-01
UR - https://www.deepdyve.com/lp/oxford-university-press/modeling-the-determinants-of-mechanical-advantage-during-jumping-OBMUYPQAnm
SP - 1515
VL - 59
IS - 6
DP - DeepDyve
ER -