Spider Origami: Folding Principle of Jumping Spider Leg Joints for Bioinspired Fluidic Actuators

Abstract Jumping spiders (Phidippus regius) are known for their ability to traverse various terrains and have targeted jumps within the fraction of a second to catch flying preys. Different from humans and insects, spiders use muscles to flex their legs, and hydraulic actuation for extension. By pressurizing their inner body fluid, they can achieve fast leg extensions for running and jumping. Here, the working principle of the articular membrane covering the spider leg joint pit is investigated. This membrane is highly involved in walking, grasping, and jumping motions. Hardness and stiffness of the articular membrane is studied using nanoindentation tests and preparation methods for scanning electron microscopy and histology are developed to give detailed information about the inner and outer structure of the leg joint and its membrane. Inspired by the stroller umbrella‐like folding mechanism of the articular membrane, a robust thermoplastic polyurethane‐based rotary semifluidic actuator is demonstrated, which shows increased durability, achieves working angles over 120°, produces high torques which allows lifts over 100 times of its own weight and jumping abilities. The developed actuator can be used for future grasping tasks, safe human–robot interactions and multilocomotion ground robot applications, and it can shed light into spider locomotion‐related questions.

. Statistical analysis of the nanoindentation measurements. Table S2. Nanoindentation test parameters.

S1. Biological details of arthropods
The group (taxon) of Arthropods evolved around 550 million years ago and includes invertebrates with an outer shell, the exoskeleton, as insects, crustaceans and arachnids, covering around 80% of all known living animals. The fundamental material of the exoskeleton is chitin, a polysaccharide similar to cellulose. Layers of chitin microfibers are held together by different proteins to form the exoskeleton and depending on the number and arrangement (hierarchical structure) of these layers and the type of proteins, the material properties of the exoskeleton can vary from flexible and soft as in caterpillars to stiff and strong as in beetles. The exoskeleton (cuticle) of arthropods can be divided into three distinct parts: the epi-, exo-, and endocuticle. The exocuticle is often highly sclerotized (stiffened) and can be followed by an intermediate between exo-and endocuticle, the mesocuticle. The endocuticle forms the innermost part of the exoskeleton, containing non-sclerotized soft microfiber-chitin-proteinlayers. It plays an important role in the growth of arthropods during the formation of a new exoskeleton for molding.
As different from vertebrates, arthropods have an open circulatory system. In a closedcirculatory system as in humans, the heart pumps the blood along defined capillaries supplying the body and its organs with oxygen. Interstitial fluid filling the space between blood vessels and cells forming their own circulation, the so-called lymphatic system, exchanging and providing nutrients such as sugar and salt, hormones and enzymes. In arthropods, there is no distinction between blood and interstitial fluid. The body fluid (hemolymph) fills up all cavities (hemocoel) inside the body and is pumped by the heart through the body. [1] Spiders (Araneae) form the largest group among the class of arachnids, which also includes other eight-legged invertebrates, such as harvestmen, scorpions and mites. Spiders consists of two body parts, the prosoma and abdomen, which are linked together by a thin connection tube (petioles). The abdomen, also called opisthosoma, is a soft inflated bag containing the heart. The front body part (prosoma) is formed by two shells, the lower shell (sternum) and the upper shell (carapax). The eight legs are connected in between these two shells. Each leg is divided into seven segments ( Figure 1B) and contains around 30 different muscles. [2] The leg can be broken down into three main joints. [3] Two of these main joints (femur-patella, tibiametatarsus), crucially involved in grasping and jumping, are lacking extensor muscles. Parry and Brown showed that the pressure inside the legs of spiders increases from 6 kPa to 60 kPa in all legs simultaneously when the spider starts moving. [4,5] They concluded that the increase in pressure is the main driving force behind the extension of the spider leg, capable of producing torques up to 0.013 Nmm at 50 kPa. [4] Several researchers proposed and experimentally showed that this pressure is not generated by the heart but in the prosoma, presumably by muscles moving the two shells towards each other creating a volume shift. [6][7][8][9] The joint structure of the spider legs plays a key role in the adaption of the hydraulic principle into a robotic application. The working principle of the spider tibia-metatarsus joint has been described by Blickhan and Barth as a bellow-like folded, anisotropic articular membrane which stores energy when folded and inflates with the increase in pressure. Thereby, the anisotropy of the membrane avoids opposing torques by reducing the axial stress components in the joint membrane as in an isotropic membrane, pressure would result in torques counteracting the extension direction. [10] S2. Materials and Methods

S.2.1 Histology
Several fixation, embedding and staining techniques have been tested. Spiders were narcotized with carbon dioxide and attached to a petri dish with dental polymer. Legs were quickly cut off from the body and a targeted stitch through the abdomen (heart and digestive organs) killed the spider. Freeze killing or ethyl acetate as killing agent influenced the preparation results and could not be used. A petri dish with wet tissue (Ringer's solution) was prepared and the femur-patella joint was carefully stretched and hold in positions with insect needles inside. To keep this position during sample fixation and embedding, a thin insect needle was stitched through the leg segments near the hinge joint side (Fig. S1). Traditional dehydration with ethanol (30, 60, 70, 90, 96, 99%) was tried with variation in incubation time (5, 10, 30 min per step), but all fixation trials resulted in too brittle and stiff samples. Similar results were observed when fixated with glutaraldehyde and osmium. A fixation with picric acid (Bouin solution [11] ) resulted in bright yellow samples (Fig. S1C). Although this method was great for microscopic observations, the picric acid has to be washed out with ethanol after staining. Therefore, to spare the samples, Hartman's fixative (also Davidson's fixative, Sigma-Aldrich H0290) often used in vertebrate tissue fixation, was used. The fixative contains acetic acid, alcohol and formalin. Samples were fixated overnight. Embedding in epoxy (Technovit) and spurr were also tested, but microtome cutting (with (sheet) glass, razor blades and (cryo) diamond knife, 4 mm, 45°, Leica) resulted in rupture of the articular membrane due to too large cutting thickness (~2 µm), the thinness of the membrane and large differences between the stiffness of the sample and the embedding material. Paraffin embedding was therefore chosen and carried out in the standard way.

S.2.2 Jumping performance estimation
Pressure is defined as the force acting on a surface area. To estimate the load -pressure behavior of our system, a constant contact area at each stable position ( Fig. S3) was assumed, resulting in a proportional increase of load with pressure. Linear fits through the experimental data at each stable position (Fig. S3) have been carried out (Table S6). Displayed linear functions ( Figure 6A) show linear fits with a mean slope of 0.013 N/kPa, corresponding to a contact surface of 13 mm². This surface would match to the surface at the entry of the chamber, were water supply is connected to (Figure 4, Fig. S3). Further studies of this surface area have to be done in following works.
The mean slope value was used to calculate the load at 200 kPa ( Figure 6C) and the resulting work ( Figure 6D), mirrored by the sum of the area below the curve (W1, W2, W3, W4). The resulting work shows a quadratic behavior with opening distance as: . (1) As the work is the integral of force over distance, a linear force-distance behavior, would explain this quadratic function and shows that the initial flexed position s of the legs has a great impact on the jumping performance. This formula was used to estimate the jumping height, whereby the work created by the elastic material was subtracted as it is needed to just lift the platform. The elastic work for four legs lies between 0.17 Nm to 0.42 Nm for legs flexed between position 75° and 100°. For calculation purposes, the mean-value (0.30 Nm) was used. The work left after deduction was treated as potential energy of the system, to estimate the jumping height as: . (2)

S.2.3 Material behavior comparison
The chosen TPU filament shows a Shore Hardness of 94A, which corresponds to a Young's modulus E of approximately 0.005GPa (4.8 MPa) from (3) For comparison, to achieve the same displacement s when bending two beams, one with the measured stiffness of spider membrane (5 GPa) and one with the stiffness of TPU (5 MPa), keeping length L, width b and applied force F of the beams the same, the thickness d of the TPU beam has to be 10 times higher than the spider membrane, computed from: . (4) Assuming the measured stiffness for the articular membrane could be 10 times lower under hydrated condition, it would still result in a ratio of 5. This means, to fabricate a membrane similar in the length scale of jumping spiders, a thickness of 250 µm of TPU membrane is needed to be comparable in its bending behavior to a 50 µm thick spider membrane. Jumping spiders Phidippus regius were bred for experimental studies. Photo shows female (white) and male (black) adult spider after copulation (90min) in the female's "sleeping bag". Male was "chased out of the house" shortly after taking the photo. Female abdomen increased in size drastically and the mother-to-be formed a dense net inside her "sleeping" bag, where juvenile spiders developed within a couple of weeks (A). For SEM preparation the dorsal side of the spider was attached to a thin metal sheet. Hot glue was used to lift the legs into horizontal position. Small hot glue "hand cuffs" were formed around the feet to create a mechanical interlocking (B). First histological experiments were tested with a picric acid solution, resulting in bright yellow paraffin cuts, good for microscopical observation, but not for staining purposes (C). To avoid flexion of the femur-patella joint during preparation and cutting, thin insect needles were stuck through the leg segment resulting in small spider leg "resistors" or "BBQ sticks". To avoid damage on the membrane side, needles were stuck through near the hinge joint side (D). For narcotization, spiders were transferred into a falcon tube and a balloon was filled with carbon dioxide (E). Nanoindentation of articular membrane (lower left) was conducted with a Berkovich tip (upper left) on aluminum substrate (F). 3D model of spider-joint-inspired actuator prototype is printed in vertical direction (femur to patella) to avoid support material inside the chamber (G). Bowden extruder head was rebuilt to direct extrusion, allowing better 3D printing of flexible material as rubber material show discontinued material pushing due to its elasticity properties (H). Experimental set-up for prototype characterization includes, the prototype with red tracking markers (1), a rotating flexion mechanism (2), a force sensor (3), a pressure sensor (4), a connection for a syringe to increase pressure (5), an Arduino Uno for sensor processing (6), a computer for recording (7), a remote control for changing into calibration mode (8) and a camera, recording folding angle (9) (I). The flexion mechanism consists of an inner latex bladder that inflates and collects the water, when the leg is flexed (J).  Fully extended femurpatella joints can also be seen in the highspeed recordings (movie S1) of spider jumps (B). To measure the pressure-load-angle behavior at the stable positions, the prototype was clamped between the two plates to a given angle. The bigger plate is attached to a load sensor, the leg to a syringe and pressure sensor. Sensor data were recorded with an Arduino Uno, while pressure was increased with the syringe. Scale bar: 10 mm (C). Table S1. Statistical analysis of the nanoindentation measurements. The distribution of Young's Modulus E and Hardness H experimental data were analyzed, p-values of the statistical tests are listed. Statistical testing was conducted with R. Normal distribution is tested with the Shapiro-Wilk-Test and the F-Test compares variances of two normal distributed groups. Pvalues above the significance level of 0.05 in these two tests are marked with an asterisk, indicating normal distribution or no difference in variance. Biological samples are often considered as normal distributed with similar variance, even when experimental data show different. As both, Shapiro-and F-Test are not clearly supporting normal distribution and similar variances, a standard t-test cannot be used to test similarity in means. Therefore, the Welch's t-Test is used, which considers differences in variances and the Wilcoxon-Test compares groups without assuming normal distribution. P-values, below the significance level 0.05 are marked with an asterisk, indicating a difference in means of the dot and line area.  Table S2. Nanoindentation test parameters. Parameters used for Nanoindenter (XP, Keysight) and data analysis for hardness and Young's modulus. Poisson ratio was chosen according to literature. [12] Depth limit corresponds to 2-10% of sample thickness to avoid substrate effects. [13] Parameter  to unsealed area • Peeling off can happen when pressurizing • Can influence "collapsibility" behavior due to thickness on the outside Filling chamber and baking at 50° for 30 min • Non-sticky inner layer, thin sheet of latex • Holes can be "healed", by repeating the process • No influence of the folding properties