A physical model of mantis shrimp for exploring the dynamics of ultrafast systems

Abstract

Efficient and effective generation of high-acceleration movement in biology requires a process to control energy flow and amplify mechanical power from power density-limited muscle. Until recently, this ability was exclusive to ultrafast, small organisms, and this process was largely ascribed to the high mechanical power density of small elastic recoil mechanisms. In several ultrafast organisms, linkages suddenly initiate rotation when they overcenter and reverse torque; this process mediates the release of stored elastic energy and enhances the mechanical power output of extremely fast, spring-actuated systems. Here we report the discovery of linkage dynamics and geometric latching that reveals how organisms and synthetic systems generate extremely high-acceleration, short-duration movements. Through synergistic analyses of mantis shrimp strikes, a synthetic mantis shrimp robot, and a dynamic mathematical model, we discover that linkages can exhibit distinct dynamic phases that control energy transfer from stored elastic energy to ultrafast movement. These design principles are embodied in a 1.5-g mantis shrimp scale mechanism capable of striking velocities over 26 m [Formula: see text] in air and 5 m [Formula: see text] in water. The physical, mathematical, and biological datasets establish latching mechanics with four temporal phases and identify a nondimensional performance metric to analyze potential energy transfer. These temporal phases enable control of an extreme cascade of mechanical power amplification. Linkage dynamics and temporal phase characteristics are easily adjusted through linkage design in robotic and mathematical systems and provide a framework to understand the function of linkages and latches in biological systems.

Keywords: bioinspired mechanisms; bioinspired robotics; linkage dynamics; mantis shrimp; ultrafast motions.

Fig. 1.

An overview of biologically inspired physical models that generate extreme accelerations. (A) A diagram illustrating high acceleration within biological and synthetic LaMSA systems. From left to right, two synthetic systems, water strider-inspired robot (44) and flea-inspired robot (69), and two biological systems, flea (70) and snipefish (36, 71), are shown. A survey of more acceleration data of biological and synthetic LaMSA systems can be found in table 1 of ref. . Water strider–inspired robot image from ref. . Reprinted with permission from American Association for the Advancement of Science. Flea-inspired robot image ©2012 Institute of Electrical and Electronics Engineers; reprinted, with permission, from ref. . Flea image credit: CanStockPhoto/ottoflick. Snipefish image credit: Wikimedia Commons/Tony Ayling. (B) Photograph of our mantis shrimp–inspired mechanism and photograph of a peacock mantis shrimp by Roy Caldwell. The proposed mantis shrimp robot generates $10^4$ m ${\text{s}}^{-2}$ for striking the arm, and the mantis shrimp generates $2.5\times 10^5$ m ${\text{s}}^{-2}$ for striking the appendage (19). Photographs adjusted for contrast with background removed. Adapted with permission from ref. . (C) (Right) The four-bar linkage in the mantis shrimp appendage is labeled (a to d). Adapted with permission from ref. . The striking arm has three tightly coupled components (dactyl, propodus, and carpus), which are colored purple. Two exoskeleton elastic components are colored blue. Last, the extensor muscle, which actuates the striking motion, is colored red. (Middle) A geometric abstraction of the four-bar linkage with two rigid bodies, the arm and the body. (Left) The synthetic realization of the proposed four-bar linkage with one variable-length link. The body is highlighted orange, and the arm is purple. Flexures which allow articulation are shown in yellow. The mechanism is secured to a 3D printed base using two screws. A tendon, shown in red, is used to actuate the mechanism. A series of holes in the base allow the tendon pulling angle to be adjusted between experiments. Potential energy is stored in a torsion spring (blue).

Fig. 2.

A planar model for the four-bar linkage of the mantis shrimp. (A) Dimensions and inertial components of two rotating bodies composed with the four-bar links ($L_0,L_1,L_2,l_t$). Arm and body are shown. The arm rotates away from the body (${\theta }_2$) as the spring recoils (${\theta }_1$). An external force, $F_t$, acts on the tendon, and a torsional spring, with spring coefficient $k_s$, is attached between the body and ground (shown here as a linear spring for convenience; a torsional spring is used in the physical system). The two generalized coordinates are ${\theta }_1$ and ${\theta }_2$. (B) Configurations before and after overcentering are shown. The tendon links, $l_t$, for both configurations are colinear and thus overlap in this drawing. (C) Direction of the generalized constraint torque, $\tau$, between the arm and body when in contact. The constraint torque is a reaction force which is nonzero only when the arm is in contact with the body. In our physical model, there is an offset contact angle, denoted as $\varphi$, between the arm and the body when they are in contact.

Fig. 3.

Temporal phase transition diagrams with representative experimental data and a comparison with biological data from mantis shrimp strikes presented in ref. . (A–C) Experimental data of the motion of the physical model, with a controlled tendon pulling speed of 2.3 mm ${\text{s}}^{-1}$ in air (Materials and Methods). (D–F) Two fast phases (phase III and phase IV) of a mantis shrimp strike (from ref. 33) are shown. (A) Snapshots of the experiments for each phase are overlaid with shadows. The last snapshot images at the end of the phase (before switching to the next phase) are displayed with a darker color. The full motion video can be found in Movie S1. (B) The filtered rotational velocities (Data Processing) of the generalized coordinate for the slow phases (phases I and II) (Left) and the fast phases (phases III and IV) (Right). Slow phase data in Left was additionally spline fitted using a free-knot spline approximation. (C) The temporal phases with representative kinematic diagrams corresponding to the video frames in A. The purple, green, blue, and brown colors represent phases I, II, III, and IV, respectively. (D) Snapshots of the mantis shrimp striking motion. (E) Filtered rotational velocity of the coordinates defined in ref. . (F) A geometric depiction based on the video frames in D of the appendage configuration of the mantis shrimp during phases III and IV with approximate linkage configurations. Adapted with permission from ref. .

Fig. 4.

Various potential energy levels achieved by different tendon pulling speeds (${\dot{l}}_t$), contact angles ($\varphi$), and joint-to-tendon lengths ($L_0$). The spring potential energy is calculated based on a linear torsional spring model with the measured ${\theta }_1$ angle data (SI Appendix, Simulation and System Identification). Higher maximum loaded potential energy, ${\text{PE}}_{max}$, can be achieved by reducing the contact angle, $\varphi$, and joint-to-tendon length, $L_0$. Also, a higher loaded potential energy at the overcentering position tends to attain a higher ${\text{PE}}_{max}$. (A) The relationship between ${\text{PE}}_{max}$ and the contact angle $\varphi$ for different pulling speeds ${\dot{l}}_t$ are shown, with the pulling speed color coded. The circles and crosses represent the $L_0=23\hspace{0.17em}\text{mm}$ tests in air and water, respectively. The triangles represents the $L_0=19\hspace{0.17em}\text{mm}$ tests in air. (B) Trajectories for phases II, III, and IV are shown in the (${\theta }_1,{\theta }_2$) space for $L_0=23\hspace{0.17em}\text{mm}$ group (in air). The circle represents the overcentering position. The trajectories below the dashed black line represent phases I and II, and the trajectories above the dashed line correspond to phases III and IV. After overcentering, the system continues to load more potential energy (reducing ${\theta }_1$), while the arm starts to rapidly rotate (increasing ${\theta }_2$). (C) Controlled contact angle tests with $\varphi =\mathrm{0,6.4}°$, and $11.4°$ showing the correlation between ${\text{PE}}_{max}$ and contact angle. The tendon pulling speeds for all seven tests are regulated to 2.3 mm ${\text{s}}^{-1}$. (D) Same as B but for the controlled contact angle experiments. (E) Analytical overcentering configurations for varying joint-to-tendon lengths, $L_0$, showing that different overcentering configurations can be achieved based on different values of $L_0$ (SI Appendix, Geometric Conditions for Over-Centering). Two reference lines show the physical limits of the system. The reference line at ${\theta }_2=0°$ represents contact of the arm and body. The second reference line represents when the body touches the base. (F) Full trajectories of experiments with the $L_0=23\hspace{0.17em}\text{mm}$ group (in air) are projected on $\left({\theta }_1,{\theta }_2\right)$. Each trajectory starts from a small ${\theta }_2$ value and a large ${\theta }_1$. The same color map as in B is used to represent ${\text{PE}}_{max}$. The surface of tendon lengths, $l_t\left(q\right)$, vs. the generalized coordinates is color coded (black to white). The black contour lines represent the level sets of $l_t\left(q\right)$, and the tendon lengths along the black contours remain constant. As the tendon is pulled, the tendon length is reduced and makes a sharp turn near the maximum spring contraction (i.e., minimum ${\theta }_1$).

Fig. 5.

Analysis of the maximum tip velocity and kinetic energy distribution during phase IV for different loading conditions (i.e., air and water) and different values of $L_0$. (A and B) The relationship between the two performance metrics and the maximum loaded potential energy, ${\text{PE}}_{max}$, is shown. Greater maximum loaded potential energy, ${\text{PE}}_{max}$, leads to higher performance for experiments in air with $L_0=$ 23 mm. (C–H) The trajectory of the maximum tip velocity and the kinetic energy distribution during phase IV versus the potential energy. The value $\text{PE}/{\text{PE}}_{max}$ describes the normalized potential energy. The filled circles represent the moment of maximum tip velocity. (A) A linear regression using the minimum mean squared error for experiments in air with $L_0=23$ mm is shown as a dashed line with ${\text{R}}^2=0.6436$. Among the $L_0=23\hspace{0.17em}\text{mm}$ group, the SD (from trial to trial) is significantly reduced in water (0.17 m ${\text{s}}^{-1}$) compared to air (2.04 m ${\text{s}}^{-1}$). (B) A linear regression using the minimum mean squared error for experiments in air with $L_0=23$ mm is shown as a dashed line with ${\text{R}}^2=0.8865$. (C) The applied force (color coded) over the course of the trajectory was less than 18 N. During the slow motions in phases I and II, the tip velocity remains nearly zero while the spring potential energy increases. Once the maximum potential energy is reached, the tip velocity rapidly increases and reaches its maximum (black circles). (D) Phase IV arm kinetic energy ratio vs. normalized potential energy for 32 experiments (in air with $L_0=$23 mm). Phase IV starts at unity for both axes (Materials and Methods). During phase IV, the spring releases potential energy, which results in a decrease of $\text{PE}/{\text{PE}}_{max}$. Potential energy is converted to kinetic energy of the arm and the body, where the ${\text{KE}}_2/\text{KE}$ metric shows how the kinetic energy is distributed between the arm and the body. The color scheme is based on ${\text{PE}}_{max}$. (E and F) Similar to the experiments in C and D but in water. The added mass effect is considered in the calculation of the kinetic energy (Materials and Methods). The fluidic loading in water causes the arm to rotate slower compare to experiments in air, but distributing more kinetic energy to the arm. (G and H) Similar to the experiments in air in C and D but with $L_0=19\hspace{0.17em}\text{mm}$. A shorter $L_0$ leads to load more potential energy, but the maximum tip velocity is not substantially increased. However, the potential energy was distributed to a greater extent to the arm compared with the experiments with a longer $L_0$.