Shape, Size, and Fabrication Effects in 3D Printed Granular Jamming Grippers

Jamming occupies a niche in soft actuation where comparatively rapid response times ($\approx$1s) and large stiffness variations are required [2]. Jamming can be categorised into three types depending on the mechanism that induces stiffness variation - granular jamming (inter-particle forces on compression), layer jamming [8] (forces between stacked layers of material), and fibre jamming [9](interactions between fibres). The three mechanisms have different strengths and weaknesses; see [3] for a recent comprehensive review.

Granular jamming is the most prevalent, with a diverse array of actuator configurations allowing for locomotion of worm-like and snake-like robots [10, 11], soft robotic ’paws’ [12, 6], and minimally-invasive surgery tools [13].

The literature evidences an increasing variety of gripper designs, from fingers and hands to hybrid jamming/pneumatic actuators [14], primarily facilitated through 3D printing of moulds for subsequent casting, or more recently direct 3D printing [15, 16]. An increasing range of actuation techniques are similarly observed, including positive pressure [5] and fluidisation [17], to jamming based on geometric confinement [18, 19] (e.g. by pressurising and expanding a neighbouring cavity), in addition to the standard negative pressure technique.

Although these more complex designs are having a substantial impact in their relevant domains, the most popular variant remains the bag-style ’universal’ gripper [4, 20], which has been combined with learning for object-specific gripping strategies [21] and found use in diverse range of application areas, including sub-sea sampling [22], and prosthetic end-effectors [23], .

Naturally-occurring grains such as coffee have implicit uncontrolled variations in shape and size distribution. Control over shape allows for bespoke grain properties, and can be realised straightforwardly through modelling techniques (predominantly DEM), and more recently via 3D printing.

Grain shape determines fundamental bulk properties including stiffness and stress responses [24], as well as packing fraction [25]. Jamming structures based on a bonded sphere representation have been optimised via genetic algorithm  [26, 27], with evolution able to create a small library of grains across the continuum of the fundamental variable under consideration. Grains were printed for experimental confirmation of results, although some instantiations of the representation were unprintable. Later work used multi-objective evolution to explore a range of bulk properties, including packing fraction, with represented by parameterised superquadrics [28]. Superquadrics are inherently printable, and thus a strong candidate representation. We note that all of the above work is carried out in the context of granular physics, and lacks direct applicability to soft robotics.

In a robotics context, information on grain performance is more scarce. We know that performance is task-dependent - work on compliant paws, for example, tends to favour cubes [12] due to their tendency to geometrically jam. Damping effects of variations of 3D printed grains with various filleting patterns highlights the flexibility of 3D printing to precisely control grain fabrication, in this case for vibration damping, however only spheres were considered [7].

Although the impact of membrane material in bag-style grippers has been studied in-depth [29], there is no comprehensive study covering applied testing of shape effects of 3D printed grains for jamming actuators. We are therefore motivated to study a diverse range of grain shapes and sizes in the context of jamming grippers, where we can easily study and precisely fabricate any desired shape. So far, only a very small subset of possible grain geometries (spheres and cubes) have been studied for soft robotics applications.

We specifically note the novelty and applicability of the superquadric formulation in this context for generating guaranteed-printable grains. Moreover, the superquadric formulation allows access to a range of ellipsoidal grain types which have found significant uptake in other fields [25], and may be straightforwardly transferable to the soft robotics domain. These shapes are not easy to purchase and typically require printing.

We chose four different grain shapes from the family of possible superellipsoids, that have been shown from DEM simulations to maximise the diversity of behaviours of the granular material in the jammed state [25, 30] (Fig. 2). Grains are parameterised following:

The first shape is a sphere ($m$=2, $a$=$b$=$c$=1), commonly employed in both simulation and experiments. Spheres lack the additional degrees of freedom of a non-spherical shape and have a highly distinct jamming density and lower average contact number compared to other superellipsoidal shapes. To incorporate the first anisotropic effects of aspect ratio, we use a prolate ellipsoid with two equal axes ($m$=2, $a$=1, $b$=$c$=0.65). This shape introduces a rotational degree of freedom and a significantly higher jamming packing density and average contact number compared to spheres.

Flat surfaces have a strong impact on the degree of orientational ordering that readily forms within a jammed system and creates unique face-on-face contacts that strongly oppose both rotation and lateral movement. To study this effect we utilise an equiaxed superball with a shape factor (exponent in the standard superellipsoid formula) of $m$=5, $a$=$b$=$c$=1. This shape is similar to a cube with rounded corners and can readily produce jammed packings with high degrees of orientational ordering along the 3 primary axes of the grains (cubatic ordering). Finally we select a triaxial superellipsoid with a shape factor of $m=3$ and aspect ratios of $a$=1 $b$=0.75 $c$=0.6, giving a cuboidal shape with highly smoothed corners. This shape sits at the intermediate points of surface curvature and angularity, with three distinct aspect ratios designed to provide a moderate amount of face-on-face contacts, while disrupting the formation of long-range orientational ordering. This set includes two popular shapes in the literature (spheres and cubes (superballs)), and two that are highly underexplored (ellipsoids and superellipsoids).

Grains are printed on the Stratasys Connex3 OBJET500 polyjet printer in the rigid vero material at 16 micron layer height. Printing time was independent of grain shape but varied slightly due to grain size; 1000 3mm grains took $\approx$ 53 minutes while 1000 7mm grains took $\approx$ 1 hour. Once printed, the grains were placed into a 1mm aperture laboratory sieve and placed in a Stratasys CSIIP CleanStation support removal bath for 24 hours. For consistency, grains at each quoted ”size” have equivalent volume to a sphere of that size, which we call Sphere-Volume Equivalent (SVE). For example, a 5SVE superball is a superball scaled such that its volume is equal to a 5mm diameter sphere.

A thin, 12.5cm latex balloon was chosen to reduce effects of the membrane on performance. The balloon was marked with a horizontal line, just below the neck, and filled up to the mark with the chosen grain type using a standard funnel. The balloon was attached to a 3D printed adapter and mounted onto a drill press stand to constrain the range of motion to vertical only, aiding repeatability. A Thomas 107CDC20-H vacuum pump was connected to the balloon gripper via 5.5mm silicone tubing.

Four test objects (a ball, cube, coin, and star) were chosen to provide diverse gripping challenges whilst being easily integrated into the test rig. The test objects were all of approximately 20mm³ in size, chosen so as to be in the range of 3x to 7x the size of the particles used in the gripper so as to maximise the size and shape variations measured in the experiment. Test objects were 3D printed with a thread and screwed into a Zemic H3-C3 load cell, which was clamped and centered inline with the balloon adapter. A flat platform (50mm x 50mm) was attached between the object and load cell to replicate the action of picking an object from a flat surface.

Testing involved lowering the gripper onto the object in an unjammed state. The vacuum was activated to cause jamming, and the gripper slowly raised until it released and completely cleared the test object. The vacuum was deactivated and the gripper manually reset by shaking. Fig. 3 the testing procedure. Each grain shape/size combination was tested 5 times per test object and mean performance reported.

For the test objects used, we see a general trend of difficulty: coin is most difficult (0.967N mean across all tests) (Fig. 4(c)), followed by ball (2.553N) (Fig. 4(a)), cube (4.889N) (Fig. 4(b)), and star (12.05N) (Fig. 4(d)). It is known from the literature that coins are difficult targets for bag-style jamming grippers [5]. Interestingly, we see 3SVE grains (sphere and ellipsoid) allow for reasonable coin grips without use of fluidization, with a very large drop off in the gripping strength for the 5mm and 7mm grains. This may be due to the key dimension of the coin (it’s height of 2.5 mm) compared to the grain axis length, where for small spheres/ellipsoids the greater ability for the grains to flow and surround the coin improves the gripping strength, while for larger grains the coin effectively sits below their half axis length, greatly reducing gripping strength.

For the cube object, there is a general trend of decreasing grip strength with increasing particle size across all particle shapes, with quite consistent gripping strengths across the different particle shapes. This consistency is likely due to the simplicity of the cube shape, with the grip being primarily driven by contacts along a set of perfectly flat planes with variations seen being being due to the greater ability of the smaller particles to conform to the cube faces.

The star is a particularly interesting shape owing to its non-convexity, with spikes that can become embedded within the jammed gripper structure, allowing it to be supported in multiple places from below. This creates significantly higher gripping strengths for the star compared to the other objects across all particle shapes and sizes, with the complexity of the interactions of the spikes with the individual particles (which have a similar length scale) leading to strong non-linear variations in the gripping performance over the set of particles considered.

The ball object presents a uniformly curved surface to the gripper that can be particularly difficult to grip in cases where the gripper fails to deform around the shape below it’s medial axis. The complexity of interaction with the curved surface is evident in the data, with high variability in the gripping performance found as particle size and shape is varied.

The ellipsoid is the best performing grain across all test objects and sizes (5.619N), then sphere (5.404N), superellipsoid (4.936N), and superball (4.5N). This suggests increased gripping performance for the target objects considered where the particles have highly curved surfaces (spheres and ellipsoids) vs a more a angular shape with flatter faces (superballs and superllipsoids). The curved surfaces permit the particles to flow more easily around the target object, allowing the gripper to better deform around the object and achieve a tighter grip. The increased gripping strength of the ellipsoid vs the sphere and the superellipsoid vs the superball also suggests that a degree of anisotropy in the particle shape increases the grip strength, which could be attributable to the higher density and greater number of particle contacts that these grain shapes have in the jammed state [25].

The data clearly demonstrates a strong interaction effect between the target object shape, the particle shapes within the gripper and the particle sizes, showing clearly that there is no single optimal shape for gripping a given object across all particle sizes. It is interesting to note that the more unconventional particle shapes (ellipsoids and superellipsoids) have superior performance to the more traditional shapes (spheres and cubes) in several cases; a promising indication that further exploration may yield even more interesting shape-performance combinations.