Geometric latches enable tuning of ultrafast, spring-propelled movements

Many organisms and engineered devices use systems of latches and springs to perform extremely rapid motions, including the impressive jumps of fleas and locusts (Bennet-Clark and Lucey, 1967; Bennet-Clark, 1975), shell-smashing strikes of mantis shrimp (Patek et al., 2007; Patek, 2019; Steinhardt et al., 2021) and many robotic jumpers (Armour et al., 2007; Aguilar et al., 2012; Ribak, 2020; Zhang et al., 2020). These systems use a class of mechanisms termed latch-mediated spring actuation (LaMSA) (Longo et al., 2019). LaMSA systems store energy and actuate movement with springs mediated by latch mechanisms (Gronenberg, 1996; Patek et al., 2011; Sakes et al., 2016; Ilton et al., 2018; Longo et al., 2019). LaMSA can yield high mass-specific power outputs and remarkable accelerations. Assuming switch-like (on–off) latches, tuning of LaMSA outputs should only occur during spring loading (Galantis and Woledge, 2003; Kagaya and Patek, 2016; Roberts, 2019). However, recently modeled and biological LaMSA systems with realistic, non-idealized dynamics can adjust their movement using latch mediation (Ilton et al., 2018; Reynaga et al., 2019; Divi et al., 2020). These discoveries encourage new emphasis on the ability to vary (i.e. tunability) of ultrafast systems – especially through latch mechanisms – for a variety of potential performance benefits (Ilton et al., 2018; Abbott et al., 2019; Bolmin et al., 2019; Divi et al., 2020; Bolmin et al., 2021; Steinhardt et al., 2021).

Latches separate and energetically mediate spring loading and spring propulsion in LaMSA systems. Latches and their associated dynamics modify forces, moment arms, geometric configurations and other internal system parameters to mediate the timing or magnitude of energy flow. Perhaps owing to the historical assumption of instantaneous latch dynamics (i.e. latches as on–off switches), coupled with the challenges of imaging these extremely fast, small and brief movements in living organisms, only recently have latch dynamics been analyzed in depth (Hyun et al., 2023; Ilton et al., 2018; Bolmin et al., 2019; 2021; Steinhardt et al., 2021). For example, an integrative engineering and biomechanics analysis of contact latches (latches operating through frictional, contact-based interactions of two structures) based on Dracula snap-jaw ants (Mystrium camillae) revealed that latch shape and removal speed mediate both the timing and magnitude of energy flow to influence output performance (Divi et al., 2020). Latches are central to mediating the flow of energy through these systems, especially for achieving cascading mechanical power amplification alongside repeated use (Ilton et al., 2018; Longo et al., 2019; Patek, 2019; Divi et al., 2020; Steinhardt et al., 2021; Hyun et al., 2023). Here, we tested the role of latch dynamics in reducing or enabling kinematic variation in LaMSA systems, specifically in terms of the role of geometry in the dynamics of the torque reversal latches found in snapping shrimp.

Torque reversal latches (also termed over-center latches) in LaMSA systems are a type of geometric latch in which the sign of the torque reverses when a rotating system transitions from spring loaded and latched to unlatched and spring propelled (Fig. 1) (Bennet-Clark and Lucey, 1967; Heitler, 1974; Gorb, 2004; Burrows, 2009; Koh et al., 2015; Kaji et al., 2018; Longo et al., 2018; Patek and Longo, 2018; Longo et al., 2019; Kallal et al., 2021; Steinhardt et al., 2021). Torque reversal latches can thus directly mediate the transformation of elastic potential energy to kinetic energy in LaMSA systems (Steinhardt et al., 2021). Torque reversal also serves as a mechanism enabling organisms to switch between muscle-driven and spring-driven movement (Burrows and Morris, 2003; Steinhardt et al., 2021). Here, we focused on a torque reversal mechanism found in some snapping shrimp species in the genus Alpheus (Fig. 1) (Ritzmann, 1974; Kaji et al., 2018; Chow et al., 2021). The torque reversal latching mechanism of Alpheus heterochaelis is hypothesized to operate as follows: (1) when the dactyl (i.e. the final segment of the appendage) is fully open, the geometry is such that a dactyl closer muscle loads energy into elastic elements rather than closing the dactyl, and (2) unlatching occurs when a trigger muscle shifts the geometry, suddenly reversing the sign of torque (Ritzmann, 1974).

Multiple parameters of a LaMSA torque reversal mechanism are central to understanding and testing the potential for enhancing or reducing variation in the kinematics of spring-propelled systems (Table 1). Considering Fig. 1 and its representation of the snapping shrimp mechanism, if the relative position of the spring attachment point to the pivot point results in an increasing torque on the dactyl about the pivot point in the opening direction, then elastic elements are loaded by the closer muscle. Otherwise, the dactyl rotates closed by simply contracting the closer muscle (muscle-driven actuation). When the dactyl is oriented such that the elastic mechanism is loaded, peak torque in the opening direction occurs when loading is complete. After the loading process, the unlatching process begins. During unlatching, the line of action of the force from the loaded spring is rotated toward the pivot point, which causes the torque on the dactyl about the pivot point to decrease until reaching zero. Additional energy need not be loaded into the spring during this unlatching process. When the line of actuation of the force is directly through the pivot point, the torque is equal to zero and the dactyl is unlatched. As soon as the spring is rotated past this zero-point, the torque reverses sign, and the spring applies torque to the dactyl, causing it to close as the spring recoils. Because the line of action of the spring force continues to rotate along with the rotating dactyl, the magnitude of torque continues to increase and accelerate the dactyl, resulting in rapid closure. In each of these dynamic states, small adjustments to the geometry, dynamics or energetics of the system have the potential to reduce or enhance variation of the output movement relative to the variation in stored elastic energy – extending to configurations in which torque reversal latching no longer occurs.

Although the above description of LaMSA torque reversal captures the geometric basis for the transition from a latched to an unlatched state, potentially tunable features of the dynamic process of transforming loaded elastic energy into kinetic energy may also affect the potential for enhancing or reducing variation in the kinematics of spring-propelled systems. Dactyl closing kinematics could be varied by adjusting the energy loaded into the system, such as by varying energy loading duration and rate. In this simplest case, a system would exhibit a positive correlation between elastic potential energy and kinetic energy, because kinetic energy is equal to elastic potential energy minus losses from friction and drag. Variation in kinematics can also be achieved through geometric adjustments that change the unlatching rate. The unlatching rate mediates the exchange of energy from stored elastic energy to kinetic energy. How torque evolves after torque reversal also directly affects how kinematics can be varied through the geometric and dynamic properties of the torque reversal mechanism as described above. It should be noted that many snapping shrimp also have an adhesive disk, which plays a role in latching; in A. heterochaelis, the focus of the present study, the adhesive latch is small and thought to play a minimal role in latching/unlatching (Ritzmann, 1974).

Snapping shrimp perform remarkable feats with their ultrafast LaMSA mechanism. Their rapid strikes – completed in less than 1 ms – generate such a fast water jet that cavitation forms behind it, resulting in a shooting cavitation bubble that can move at over 30 m s⁻¹ (Versluis et al., 2000; Lohse et al., 2001). Their snaps are used for a number of ecologically important functions, including communication, territorial defense, intraspecific and interspecific agonistic interactions, prey capture and even tunnelling (Winn and Hazlett, 1962; Schein, 1977; Knowlton and Keller, 1982; Beal, 1983; Vermeij, 1987; Hughes, 1996a,b, Schmitz and Herberholz, 1998; Herberholz and Schmitz, 1999; Silliman et al., 2003; Dinh et al., 2020; Dinh and Radford, 2021; Dinh, 2022; Dinh and Patek, 2022; Kingston et al., 2022). Snapping shrimp claws have been examined in the contexts of their material properties and ability to consistently and efficiently generate cavitation and plasma (Koukouvinis et al., 2017; Alam et al., 2018; Amini et al., 2018; Tang and Staack, 2019). Even with sustained attention on these remarkable animals, it remains uncertain how they store elastic energy and whether torque reversal latching is used for enhancing or reducing the variation of their strikes amidst the diverse behavioral and ecological contexts for their snaps. For instance, an increased scope for strike variation could allow snapping shrimp (or any biomechanical system) to adjust output in response to different stimuli, such as prey type. In contrast, mechanisms that reduce strike variation in response to environmental fluctuations or short-term perturbations could also be favorable.

We addressed the following questions in this study: (1) How do snapping shrimp store elastic energy and when is elastic energy stored and released relative to the timing of cocking and release of the dactyl? (2) At the scale and configuration of snapping shrimp, what are the dynamic consequences of geometric parameter combinations (in terms of dactyl starting angle and torque reversal angle) for the presence and absence of latching by torque reversal and for the enhancement or reduction of variable kinematics given varying elastic potential energy storage? (3) Based on a large experimental dataset that encompasses individual variation in snapping behavior, which realm of torque reversal dynamics modeling in question 2 is used by A. heterochaelis snapping shrimp such that their torque reversal mechanism enhances or reduces variation of their strikes?

To address these questions, we developed a reduced-order model of a LaMSA torque reversal mechanism similar to the one found in snapping shrimp (Fig. 2, Table 1). Our goal was to simulate how variable and consistent outputs can be achieved through modifications to the torque reversal mechanism without exact predictions of performance. We probed the boundary between torque reversal configurations that can enable LaMSA or can only mediate direct muscle contractions without latch-like dynamics. We performed an experimental analysis of the relative variability of elastic energy storage and strike kinematics in live snapping shrimp. Although much of the historical focus on these systems has emphasized ultra-high speeds or accelerations, we focus here on the two parameters for these strikes: dactyl rotation and strike duration. This approach keeps the same units for loading, unlatching and striking, which is necessary for comparisons of variation between these phases of the mechanism. We tested whether snapping shrimp follow the predictions for enhancing or reducing the variation of their strikes relative to their variation in energy loading using simulations with their particular torque reversal configuration. Additionally, we unexpectedly documented a previously unrecognized mechanism for elastic energy storage in snapping shrimp, specifically deformation of the propodus exoskeleton during loading and recoil.

Bigclaw snapping shrimp (Alpheus heterochaelis Say 1818) were collected at low tide from oyster reef habitats at the Duke University Marine Lab in Beaufort, NC, USA (NCDENR Scientific and Education permit 707075 to Duke Marine Lab). Filming took place at the Duke University Marine Laboratory or after transport back to the Patek laboratory at the main campus of Duke University in Durham, NC, USA. Individuals were housed in 10.16 cm³ tanks and received partial water changes and frozen krill three times a week. As detailed below, snapping shrimp were filmed during spring loading and snapping. These behaviors were elicited by placing a metal probe or paintbrush near each individual.

To parameterize the mathematical model with mass, density and length dimensions similar to those of snapping shrimp, we measured snapping shrimp molts using images from a microscope (165M FC microscope; DFC 450 C camera, Leica Microsystems, IL, USA). Based on these measurements, we represented the mathematical model's dactyl as a cylinder with a length of 5.756 mm and width of 1.025 mm. The mass of the modeled dactyl was estimated as 4.750 mg based on the cylinder's volume and water density.

With an additional series of morphological measurements, we further populated the mathematical model with parameters that match the snapping shrimp's geometry and dynamics. We validated prior morphological research (Ritzmann, 1974; Kaji et al., 2018) by visualizing the relative locations of the apodeme (i.e. arthropod tendon) insertion and joint location using dissections and dye-enhanced micro-computed tomography (micro-CT) scans of fixed claws and molts (Fig. 1; Fig. S1) (XT H 225 ST scanner, Nikon Corp., Tokyo, Japan). The spring attachment point in the model was defined with the assumption that apodeme recoil delivers elastic energy to the dactyl as it rotates around the pivot point. Therefore, we used morphological measurements of the apodeme insertion point relative to the pivot point. Specifically, we measured the horizontal distance (a_h: 1.025 mm) and vertical distance (a_v: 0.264 mm) from the pivot point to the spring attachment point on a snapping shrimp molt (165 mol l⁻¹ FC microscope; DFC 450C camera, Leica Microsystems).

To generate the images in Fig. 1B, a claw was removed from a male specimen and immediately fixed in a 1:1 solution of Bouin's fixative and 100% ethanol for 24 h. The specimen was gradually transferred to 70% ethanol before staining in an iodine solution for 3 days (3:1 70% ethanol:Lugol's solution). The specimen was then micro-CT scanned (125 kV, 80 μA, voxel size 11.4 μm, 1518×1393×1514 voxel resolution). For the images in Fig. S1, a dactyl was obtained from a fresh molt, allowed to dry and then scanned (123 kV, 87 μA, voxel size 12.0 μm, 1213×1048×923 voxel resolution). CT Pro 3D software (Nikon Corp., Tokyo, Japan) was used to reconstruct each scan. Figure images were generated by AvizoLite 9.5.0 (Thermo Fisher Scientific, Waltham, MA, USA) (Fig. 1B) or OsiriX MD v8.5 (Pixmeo SARL, Geneva, Switzerland) (Fig. S1).

Once energy has been fully loaded in the modeled system, the unlatching process begins (red shaded area in Fig. 3). During the unlatching process, spring force rotates through the unlatching angle (θ_ul) at the unlatching rate in the direction opposite to the dactyl rotation (Figs 1 and 3). When the sum of the dactyl angle (θ) and unlatching angle (θ_ul) is equal to the torque reversal angle (θ_tr), then torque (τ) is zero (vertical dashed line in Fig. 3), because the line of action of the spring force goes through the pivot point (Fig. 1D). As soon as the spring force rotates past this point, the torque reverses sign.

We collected a small dataset to establish exoskeletal deformation in the scientific literature as an additional location of elastic energy storage in this system (previously suspected by Kaji et al., 2018), and to assess the timing of propodus exoskeleton deformation during the loading phase for informing the assumptions of our modeling. To do this, we mounted the claw in place, because the claw normally rotates substantially during a snap, making it difficult to track propodus deformation accurately. Therefore, we affixed a small portion of a toothpick to the dorsal or lateral surface of the propodus and to the posterolateral surface of the carapace with respect to the orientation of the shrimp. This setup immobilized the propodus relative to the body, while allowing the dactyl to rotate freely. The snapping shrimp's body was then attached by a second toothpick to a rig outfitted with micromanipulators so that the claw could be precisely placed within the camera's field of view. Claw deformation was then filmed in the ventral view using the camera and mirror setup described below. We filmed the entire loading phase (30,000–50,400 frames s⁻¹, 18.84–32.33 μs shutter, 896×448–1024×688 pixel resolution) and then down-sampled for the digitizing analysis given the slow rate and long duration of loading.

We measured the timing of the start of propodus deformation and percent change in deformation (1 male and 1 female; 4–25 videos per shrimp). We began measuring when the dactyl was cocked in place and ended at the frame when the dactyl first started to close (Fig. 4). We measured the change in distance between two curvatures along the medial and lateral surface of the propodus visible in the ventral view of the videos. We measured the shortest distance between the opposing curvatures for a given frame (Fig. 5A). Each loading video was thresholded in ImageJ (version 1.53f51), such that the edges of the deforming saddles were clearly outlined. Using the DLTdv8a app (version 8.2.5) (Hedrick, 2008) in MATLAB (R2021b, MathWorks, Natick, MA, USA), we measured the change in distance between two curvatures of the propodus from the lateral view starting 30–200 frames before when the dactyl was cocked in place and ending 1–31 frames after the frame when the dactyl first started to snap closed (Movie 1). We calculated the percent change in deformation as the change in length occurring between cocking and snapping divided by the starting (unloaded) length between the two curvatures of the propodus prior to cocking. We used this percent distance as a proxy for propodus deformation such that we could detect the timing of elastic energy storage; this metric does not directly measure the amount or rate of energy stored or the total distribution or location of energy storage in the propodus.

Based on this analysis of propodus deformation (see Results), we established that the duration of dactyl cocking to closing is an effective proxy for duration of elastic loading. Furthermore, measurements of dactyl cocking are feasible in unrestrained kinematics experiments (next section), whereas consistent measurements of the duration and timing of propodus deformation are not possible in less-restrained animals. Therefore, for brevity and congruity with the modeling terminology, when we henceforth refer to snapping shrimp spring loading duration, it is synonymous with the duration between dactyl cocking and first movement of the dactyl in a strike.

In a separate, much larger dataset than that for the propodus deformation analysis, we analyzed loading and snapping of 80 strikes from 8 individuals (6 males, 2 females; 5–13 strikes per individual). Prior to filming, a toothpick was attached to the dorsoposterior surface of the carapace using a cyanoacrylate adhesive (Loctite^R Super Glue Gel Control^TM) to assist with positioning during filming. We filmed snaps using a high-speed video camera focused on the animal and – in the same field of view – a mirror oriented at 45 deg to the camera's plane of view. This allowed us to simultaneously visualize two sides of the snapping claw to ensure that we could digitize the view in which the closing of the dactyl was perpendicular to the camera's view (snapping shrimp readily vary the orientation of their snapping claw relative to their body axis). We filmed snaps with a high-speed video camera (FASTCAM SA-Z type 2100K-M-64GB, Photron, San Diego, CA, USA). Videos were recorded at either 50,400 or 100,000 frames s⁻¹ (50,400 frames s⁻¹, 18.84 μs shutter, 640×520 or 896×448 pixel resolution; 100,000 frames s⁻¹, 9.00 μs shutter, 384×408 pixel resolution). These two frame rates were used because smaller individuals strike more quickly, requiring a higher frame rate than the larger individuals. Given the goal of measuring individual variation and reducing variation caused by the experimental setup, we only included videos in the analysis if the claw was oriented <20 deg from horizontal in the lateral view, the video had high contrast and resolution in the ventral view, and the dactyl and bubble were visible for the entire duration of the strike. For each individual, we only analyzed videos collected within 1 month of the first collected video. Videos were collected in October–November 2018 (5 individuals) or July 2019 (3 individuals).

We measured loading duration beginning at the frame when the dactyl is first fully open and ending with the first frame with visible movement of the dactyl closing. Strike duration was defined as the first frame during which the dactyl begins to close until the frame when the dactyl is fully closed against the propodus (Fig. 4). Given that dactyl morphology constrains the rotation angle and distance traveled by the dactyl during a strike, strike duration is negatively related to strike velocity and acceleration. We digitized points on the dactyl and propodus to determine angular displacement of the dactyl (v 2.0.0, Fiji) (Schindelin et al., 2012). We calculated error (uncertainty) ratios for durations for each individual based on the coarsest high-speed video frame rate used for that individual (Table 2). Note that loading sequences were downsampled from the frame rate used for snapping kinematics, because loading is orders of magnitude longer in duration than snapping. Consequently, loading and snapping data have different measurement uncertainty. Loading duration uncertainty ranged from 0.2% to 3%, and loading durations are reported in increments of 1, 2 or 10 ms, depending on the individual. Strike duration uncertainty ranged from 1% to 3%, and strike durations are reported in increments of 0.01 or 0.02 ms, depending on the individual.

We compared variation in loading to variation in strike duration for each individual using the coefficient of variation (CV). The CV is the ratio of standard deviation to the mean and therefore allows for comparison of dispersion when the scales (means) are different. To represent how relative variation decreases or increases between loading and strike durations, we calculated a CV ratio (CVR; strike duration CV divided by loading duration CV). We applied CVR, a novel metric, to compare relative variation in spring loading duration to relative variation in strike duration.

We tested the association between loading duration and strike duration using a linear mixed model with individual ID as a random effect (lmerTest) (Kuznetsova et al., 2017) in R v 3.6.2 (https://www.r-project.org/). We used a type III ANOVA with Satterthwaite approximation for degrees of freedom to determine significance.

In our first set of simulations, we modeled the mapping of loading duration to strike duration by varying (1) energy loading duration (Δt), (2) energy loading rate (ΔE/Δt) and (3) unlatching rate (dθ_ul/dt) (defined in Fig. 3A). The energy loading duration and energy loading rate determine the timing and magnitude of elastic energy storage, while the unlatching rate represents the change in the line of action with respect to time. Geometric parameters, such as length, width, pivot and spring locations, were held constant. We defined loading duration as the sum of energy loading duration and duration from the end of energy loading to the start of torque reversal (Fig. 3). Note that this measure of loading duration is defined through events in the model that are not easily measured or apparent in the biological system, because torque reversal is not visible externally in snapping shrimp.

To parameterize this simulation, we set the mean values of the energy loading duration, energy loading rate and unlatching rate to match the observed results of the high-speed imaging study (loading duration: 409.9 ms, strike duration: 0.7 ms). The dactyl starting angle [θ(0)] was set to 10 deg and the torque reversal angle (θ_tr) was set to 15 deg based on the biological measurements. We set standard deviations for each of these parameters such that the CV (the ratio of the standard deviation to the mean) was 20%. Therefore, when we stochastically varied the reduced-order model, each simulation set these parameters from independent draws from normal distributions (energy loading duration: µ=392 ms, σ=89.8 ms; energy loading rate: µ=15 mJ s⁻¹, σ=3 mJ s⁻¹; unlatching rate: µ=450 deg s⁻¹, σ=90.0 deg s⁻¹). Note that the mean energy loading duration is smaller than the average loading duration from the high-speed imaging study, because the numerical model measures the loading duration as the duration from the start of the energy loading to the time when the torque reverses sign. In these simulations, we individually perturbed energy loading duration, energy loading rate and unlatching rate, keeping the other two constant at their means. We then simulated the more complex case in which all three parameters could vary simultaneously: each parameter was drawn from the same normal distributions described above, remaining independent from the other two. We stochastically varied these parameters over a total of 400 simulations. Although it is possible that these parameters could be inherently linked in the organism, that scenario was not modeled here.

In a second set of simulations, we tested how the geometry of a torque reversal latch affected strike duration. We varied two parameters: (1) dactyl starting angle [θ(0)] and (2) torque reversal angle (θ_tr) (Fig. 2B,C). We simulated distinct combinations of the dactyl starting angle (from 0 to 80 deg) and torque reversal angle (from 0 to 45 deg) in 1 deg increments. First, we used the average values described above for energy loading duration, energy loading rate and unlatching rate, and then calculated the strike duration at each combination of dactyl starting angle and torque reversal angle. Then, to reduce computation time, we simulated 45 evenly spaced combinations of dactyl starting angle and torque reversal angle. We ran 27 simulations for each combination of angles [θ(0) and θ_tr] in which we varied energy loading duration, energy loading rate and unlatching rate at their respective minimum, mean or maximum values. We calculated the CV as we also did for the biological kinematic data (described above) and then compared the CV between the simulations and snapping shrimp.

All simulations were performed in MATLAB (2018a) using a forward Newton–Euler method with a time step of 1 µs. Code is available via the Dryad archive for this publication (https://doi.org/10.5061/dryad.sqv9s4n6k).

The goal of our simulations was to generally model the phenomenon of torque reversal latches at similar, generalizable scales as found in LaMSA systems such as the snapping shrimp. We did not aim to explicitly model the snapping shrimp system. Our simulation approach instead aimed to test variation in torque reversal latches to generally understand how and when variation in the kinematic output of a torque reversal latch can occur across a far wider range of variables than might be found in a single biological system.

Therefore, implicit to the model description above, we have made a series of simplifying assumptions in the model, which we outline here. We modeled the dactyl with a simple cylindrical geometry, whereas the snapping shrimp's dactyl has complex morphology that could affect hydrodynamic loading forces on a striking dactyl. We serialized energy loading and unlatching processes to enable simulation of varying inputs, but this does not replicate the likely synchronized process in live snapping shrimp. Given our new discovery of propodus deformation during loading, snapping shrimp likely store elastic energy in both the propodus exoskeleton and apodeme; however, our model assumes (based on previous hypotheses for this mechanism) that elastic energy is delivered to the dactyl solely by recoil of the apodeme at the apodeme attachment location. Therefore, it is possible that in snapping shrimp, like the recently established dual spring force couple mechanism discovered in the trap-jaw ants (Sutton et al., 2022), the output rotation is even faster given an additional displacement of the pivot by recoiling propodus exoskeleton. However, this possible revision of the elastic mechanism should not change the role of the torque reversal latch and the relative variation of the input and output durations, as we are examining in this simulation. For the same reason, shifts in the modeled stiffness of the elastic mechanism might influence the magnitude of the outputs, but it would not affect the role of the torque reversal latch, which is the focus of our simulations. Finally, we tuned the average model kinematic output to match the average kinematic output of the experimental data, which required free fitting parameters. Consequently, the modeling approach offers limited ability to exactly predict the numerical values of strike duration, velocity or acceleration, such that changing assumptions about geometry, energy loading and unlatching would therefore change the kinematic output of the model.

Propodus exoskeletal deformation began as soon as the dactyl was fully opened and continued until the onset of dactyl closure (Fig. 5). Once the dactyl closure began, the deformed propodus exoskeleton recoiled back to its original configuration.

In snapping shrimp, relative variation of strike duration was considerably lower than in loading duration (Fig. 4). Loading duration and strike duration were not correlated (F_1,71.997=2.7, P=0.104). Loading duration ranged from 160 to 616 ms (390±90 ms) (Table 2). Strike duration was much shorter, ranging from 0.44 to 1.02 ms (0.70±0.14 ms). Loading CV ranged from 0.08 to 0.30. Strike CV ranged from 0.03 to 0.11. Strike CV was less than loading CV for 7 out of 8 individuals. Overall, median strike duration CV was significantly lower than median loading duration CV (one-tailed paired Wilcoxon signed rank test: V=1, P=0.008). Seven individuals had CVRs less than 1 (0.20–0.48), indicating that relative standard deviation of the loading phase was lower by more than half compared with the strike phase. One individual had nearly identical CV values, resulting in a CVR of 1.07.

The relationship between loading duration and strike duration was tunable in both of our simulations. In the first simulation, this was achieved by varying energy loading duration, energy loading rate and unlatching rate individually and in combination. An animation of the model is shown in Movie 3.

Strike duration was less variable (CV=0.054) than loading duration (CV=0.21) when energy loading duration was varied while fixing both energy loading rate and unlatching rate at their respective means. Loading duration and strike duration CVR was 0.26. Strike duration was inversely related to loading duration (Fig. 6A), because longer loading durations, with a constant energy loading rate, corresponded to more elastic energy in the system and thus shorter strike durations. Varying energy loading rate caused variation in the amount of stored energy, such that strike duration was variable even with a constant loading duration (CV=0.042) (Fig. 6B).

Varying unlatching rate was key to tuning the relationship and relative variation between loading and striking duration. Specifically, our simulations demonstrated that when the unlatching rate was exclusively varied, the loading duration and strike duration were both affected (Fig. 6C). Loading duration included both loading duration and unlatching duration until torque reversal (Fig. 3). Varying the unlatching rate caused minimal variation in the loading duration (CV=0.0053), and more strongly influenced variation of strike duration (CV=0.022, CVR=4.1).

When energy loading duration, energy loading rate and unlatching rate were independently and simultaneously varied, loading duration and strike duration were negatively correlated (F_1,98=85.6, P=5.13e–15) (Fig. 6D). Strike duration CV (0.064) was lower than loading duration CV (0.21).

We identified a boundary between latching and non-latching systems. This boundary was revealed by varying latch geometry through simulations of dactyl starting angle [θ(0)] and torque reversal angle (θ_tr) (Fig. 7). This boundary (solid black line, Fig. 7A,B) occurred when the difference between dactyl starting angle (Fig. 2B) and torque reversal angle (Fig. 2C) was equal to zero. If the difference between the dactyl starting angle and torque reversal angle was positive, then the dactyl closed through direct actuation of the motor (muscle). Muscle-based actuation was not simulated (white spaces, Fig. 7A,B). When the difference between the dactyl starting angle and torque reversal angle was negative, then spring-actuated motion occurred through the torque reversal latch.

On the side of the boundary when the system was latching, dactyl starting angle and torque reversal angle affected the strike duration. Strike duration ranged from a minimum of 0.58 ms (dactyl starting angle, 45 deg; torque reversal angle, 45 deg) to a maximum of 0.77 ms (dactyl starting angle, 0 deg; torque reversal angle, 0 deg) (Fig. 7A). When the torque reversal angle was held constant, strike duration decreased with increasing dactyl starting angles. A shorter amount of time was needed to close the dactyl, given that the dactyl began in a more closed configuration. When the dactyl starting angle was held constant, the strike duration decreased with increasing torque reversal angle. The increase in torque reversal angle increased the magnitude of the applied torque after the torque reversal event; this caused acceleration of the dactyl and thereby decreased strike duration.

Dactyl starting angle and torque reversal angle influenced the variation in strike duration. Even when using the same set of parameters for loading duration, energy loading rate and unlatching rate, relative variation between loading duration and strike duration were tuned through dactyl starting angle and torque reversal angle. For instance, CVR decreased for dactyl starting angles and torque reversal angles closer to the boundary between latching and non-latching systems. Sets with the same difference between the dactyl starting angle and torque reversal angle had the same CVR. Greater differences between dactyl starting angle and torque reversal angle corresponded to a higher CVR. For example, a dactyl starting angle of 0 deg and torque reversal angle of 5 deg (a difference of −5 deg) had a CVR of 0.33, and a dactyl starting angle of 40 deg and torque reversal angle of 45 deg (a difference of −5 deg) also had a CVR of 0.33. A more extreme example of a simulation with a dactyl starting angle of 0 deg and torque reversal angle of 45 deg (a difference of −45 deg) yielded a CVR of 0.40. With these configurations, relatively more variation occurs in strike duration even though the respective simulations use the same set of energy loading durations, energy loading rates and unlatching rates.

The simulated ‘individual’ from Fig. 6 has a dactyl starting angle and torque reversal angle closer to the boundary between latching and non-latching systems than the rest of the simulated data. We plotted the simulation results from Fig. 6D as a single circular marker in Fig. 7B. This marker has a relatively low CVR of 0.30 (dark blue).

Compared with our snapping shrimp kinematics dataset, our simulations yielded a stereotyped strike duration (CV, 0.064; CVR, 0.30; Fig. 6D; note that these simulations are based on a different geometry than in Fig. 7). The range of loading durations from the model was similar to those observed from snapping shrimp (Fig. 6D).

Our integrated approach of collecting a large kinematic dataset from snapping shrimp and developing a reduced-order model to simulate the effects of small geometric changes to torque reversal latches revealed multiple insights about the role of geometric latches in spring-propelled systems. In snapping shrimp, the torque reversal latch plays a key role in reducing the effect of variation that occurs during spring loading on the strike kinematics. Even so, small adjustments to the torque reversal system as expressed in the model reveal a boundary between spring-driven and muscle-driven configurations, and between an ability to vary kinematic output or nearly eliminate variation in kinematic output. As we discuss below, our study offers a new lens through which to view the elastic mechanism in snapping shrimp, a quantitative comparative framework for the role of geometric latches in biology, and a new framework for using small shifts in geometry to enable control of ultrafast systems in biology and engineering.

We established that snapping shrimp deform their propodus exoskeleton during spring loading and that the exoskeleton recoils when the dactyl rotation begins. Spring loading occurs throughout the time period between dactyl cocking and release (Fig. 5). This type of exoskeletal elastic loading is consistent with mechanisms used by other arthropods with LaMSA mechanisms (Patek et al., 2004, 2007; Zack et al., 2009; Burrows and Sutton, 2012; Patek et al., 2013; Rosario and Patek, 2015; Larabee et al., 2017; Patek, 2019; Sutton et al., 2022). To our knowledge, propodus deformation as an elastic mechanism has not been previously established in snapping shrimp.

Given that the dactyl is not closing while the propodus is deforming, elastic energy is likely stored both through the visible exoskeletal deformation and stretching of the internal apodeme. Therefore, our measurements of loading duration encompass the maximum duration for elastic energy storage, as well as unlatching duration and any additional processes that occur prior to the observed onset of the striking motion (e.g. delays owing to dactyl inertia).

Consistent with the timing of propodus deformation, prior research demonstrated that activity of the main spring loading muscle (Cl₁) begins immediately after the dactyl opens, and that activity of this muscle overlaps with trigger muscle (tr) activity prior to dactyl closure (Fig. 1) (Ritzmann, 1974). Deformation of the propodus during the loading phase validates that the muscles inside the propodus are indeed active during this period; propodus shape deformation most likely occurs owing to Cl₁ muscle activity compressing the propodus exoskeleton while the dactyl is latched (although trigger muscle activity could also contribute).

The discovery of propodus deformation in snapping shrimp may point toward their use of a dual spring force couple, as was recently discovered in trap-jaw ants (Sutton et al., 2022). Dual spring force couples in LaMSA systems are formed by combination of a recoiling apodeme and deformation of exoskeleton to develop torque on rotating spring-propelled systems. Although it is not yet clear where the recoiling propodus exoskeleton would develop torque in snapping shrimp, one possibility is that the recoil pushes the dactyl's pivot distally. This would cause the dactyl to rotate even more quickly than predicted by our model (our mathematical model only incorporates apodeme recoil to power the dactyl's rotation). As addressed in the Materials and Methods, this possible new mechanism for spring propulsion in snapping shrimp is worthy of further study but should not substantially change the conclusions of the present study. Our model was developed to test the role of torque reversal on tuning the relative variability of spring loading and strike duration, and the model was not structured to predict strike velocity for a specific system.

We found that, in addition to their role in geometrically mediating the onset of movement as a latch in LaMSA systems (Steinhardt et al., 2021), torque reversal latches also can mediate the magnitude and relationship between input and output variation. Our mathematical model reveals how torque reversal latches confer substantial tunability of the correlation and relative variation between loading and strike durations. By structuring the model to serialize energy loading and unlatching processes of the torque reversal mechanism, we were able to simulate and test how independent change of each process affects loading and strike duration. Our use of a reduced-order model allows us to characterize variability in the dynamics of torque reversal latches in general, similarly to the template and anchor approach in legged locomotion (Full and Koditschek, 1999). The ability of the torque reversal latch simulated here to manipulate the relative variation between loading and strike phases has important implications for robust and repeatable biological and synthetic systems, especially considering that this form of control can be implemented mechanically through latch design (Koh et al., 2015; Steinhardt et al., 2021). The question of whether all latches (e.g. contact, fluidic) or even other geometric latches (e.g. mechanical advantage) similarly attenuate input to output variation is an exciting avenue for future research (Hyun et al., 2023). This is particularly interesting for synthetic design when considering trade-offs between the demands of high performance with output tunability.

The reduced-order modeling approach allows hypothesis testing about the torque reversal mechanism and the effects on performance when perturbing individual actuation parameters that govern elastic energy loading and energy mediation during unlatching (Fig. 3). Simultaneously and independently allowing energy loading duration, energy loading rate and unlatching rate to vary produces output with similar levels of variation as within individuals in the biological data (Fig. 6D). However, the simulated data show a slight negative correlation between loading duration and strike duration that is not seen in the biological data. Note that the simulations in Fig. 6D represent many samples from a single ‘individual’ (a fixed set of geometric parameters), while the spread of the biological samples along the y-axis is attributable in part to individual variation (e.g. animal size). Overall, the simulations suggest that variation in input actuation could account for the variability in loading duration observed in snapping shrimp. Given that muscle activity is not visible in the high-speed videos and is known to vary in other LaMSA systems (Lappin et al., 2006; Kagaya and Patek, 2016), snapping shrimp also likely vary the muscle contractions responsible for energy storage and unlatching.

Although sequential actuation in our model is insightful, it is important to note that many biological and synthetic systems may not have independent control over spring loading and triggering unlatching to allow for tunability. Moreover, the actuators (e.g. muscle) responsible for spring loading and triggering will have their own dynamics. For instance, in snapping shrimp, the trigger muscle activation overlaps with loading muscle activity (Ritzmann, 1974). Similarly, in a previously developed ‘flea robot’ that used a torque reversal latch, the actuators for energy storage and unlatching are coupled through the circuitry and thus are temporally linked (Noh et al., 2012). If any of the actuation parameters are coupled, we would expect overall less variation in both our input and output timing measurements. The pathways by which variation decreases would be a result of both how the actuation parameters are coupled, as well as their inherent actuator dynamics, where transient dynamics of actuators (e.g. muscles) would affect timing in loading and the system unlatching. For example, in our model, setting the unlatching to always start when a force threshold is reached, analogous to an adhesive latch, would result in a decrease in variation in both loading duration and strike duration. Such an adjustment would apply to other snapping shrimp species in which adhesive discs play a more significant role in latching (Ritzmann, 1973; Ritzmann, 1974).

Our simulations suggest a generalized principle of torque reversal systems such that biological systems can evolve torque reversal systems that either reduce or enhance variable outputs. Our large high-speed video dataset reveals a stark contrast between the high magnitude and variability of the loading duration and the low variation of strike duration in the snapping shrimp A. heterochaelis, without a significant relationship between loading and strike duration across individuals. The reduced-order model for the torque reversal latch allowed us to evaluate the role of the latch in producing these biological observations. We found that for a fixed set of geometric parameters similar to an individual of this species, the system can achieve a stereotyped output relative to input. Therefore, torque reversal latches can attenuate variation by their fundamental dynamics, reducing the magnitude of variation of the outputs relative to the inputs, and allowing for relatively stereotyped outputs even with perturbations during spring loading and actuation.

Our work demonstrates how subtle changes in geometry mark a transition from non-latching to latching systems with rapid, highly stereotyped output (Fig. 7). We explored the consequences of two adjustments: dactyl starting angle and the torque reversal angle, which represents the attachment point of the spring relative to the pivot point. We found that these two adjustments influence the presence/absence of a torque reversal latch, vary strike duration, and change the relative variation between energy loading duration, energy loading rate and unlatching rate to the strike duration (represented by the CVR between loading and strike duration).

When torque reversal latches are near the boundary at which a latch would no longer be present, they have the lowest CVRs. In other words, torque reversal latches with these specific parameters attenuate variation contributing to output stereotypy, while configurations with parameters farther from this boundary could have more control of strike output. Our results also demonstrate that geometric parameter pairs that are parallel to the boundary line have equal CVRs and affect the magnitude and relationship of input and output variation owing to their similar dynamics, despite differences in the magnitude of strike durations (Fig. 7B).

Seemingly small changes in dactyl morphology result in the sudden transition from a directly actuated dactyl to one with a spring-driven, torque reversal latch (Fig. 7). This transition mirrors a similar boundary for a flea-inspired robot with a torque reversal mechanism (Noh et al., 2012) and reflects similar dynamics of latch release and energy flow in mantis shrimp (Patek, 2019; Steinhardt et al., 2021). The stark boundary is characteristic of a system in which relatively small changes in certain aspects of morphology (but not others) can generate seemingly disproportionate jumps in performance (Koehl, 1996; Anderson and Patek, 2015; Muñoz et al., 2018). Torque reversal latches have evolved multiple times in snapping shrimp (Kaji et al., 2018; Chow et al., 2021), and our simulation hints at how sensitivity to small changes in joint or pivot morphology would allow organisms to transition across the non-latching to latching boundary. Indeed, joint and apodeme/tendon insertion positions that are nearly aligned may act as preadaptations for geometric latches.

The ability to vary movement in response to a specific situation or stimulus can increase performance. However, given the rapid speed and short durations of LaMSA movements (Sakes et al., 2016; Ilton et al., 2018), responding to changing stimuli can be difficult or impossible using neuromuscular feedback loops (Hyun et al., 2023). This, in turn, seems to be at odds with the fact that snapping shrimp can produce rapid, cavitation-inducing dactyl closure in diverse environmental and behavioral contexts. Here, we found that mechanical features of the underlying components play important roles in determining input-output responses. Considering the difficulty of generating cavitation in water, the torque reversal latch in snapping shrimp may mitigate small differences in loading and triggering energetics while still allowing for stereotyped, short-duration, high-energy dactyl closure to occur. Latch geometry similar to A. heterochaelis resides near the transition between latching and non-latching morphology (Fig. 7), where the attenuation of input to output variation is expected to be strongest. A morphological configuration that attenuates variation from the input to a stereotyped output could compensate for perturbations in inputs, including from changes in the muscle dynamics over repeated use or from fluctuations in the environment itself. The torque reversal latch also allows control over the onset of impulsive movement (Steinhardt et al., 2021), potentially allowing organisms to spend variable amounts of time aiming, reorienting, sensing or bracing for an imminent strike, without compromising output (Crane et al., 2018). This has important implications for trait evolution and mechanical sensitivity (Muñoz et al., 2017, 2018; Muñoz, 2019), while also demonstrating that evolutionary transitions to torque reversal latches may be favored when there is a need to be flexible in the face of uncertain conditions, while maintaining consistent output.

Latches can ultimately dictate the control authority over output performance by geometrically mediating variation and coupling across energy input and kinematic output (Hyun et al., 2023). For example, an engineered LaMSA with a torque reversal latch may be able to produce a stereotyped output under a variety of conditions from the predictable mechanics of the latch itself, which alleviates the need for ‘on-board’ sensing and processing, especially for size- and resource-constrained systems (St. Pierre and Bergbreiter, 2019). Conversely, latch geometry can attenuate variation from imperfect manufacturing or actuation, reducing a need for active control to produce a consistent output. Manufacturing tolerances can have a significant effect on mechanisms such as springs as the engineered system decreases in size (Caro et al., 2005; Dieter and Schmidt, 2012). When using springs with relatively large tolerances for actuation, a torque reversal latch with predictable mechanics presents a significant pathway for tuning, enhancing or reducing variability in pre-loaded elastic mechanisms found in both biological and engineered LaMSA systems.

We are grateful to members of our Impulsive MURI team, Patek Lab and Bergbreiter Lab for helpful feedback on this research. We also thank those who helped with animal collection, filming, animal care and digitizing, especially W. Ray, G. Farley, J. Dinh, S. Hanson, G. Pentapalli, A. Lin and M. Huang.