Soft Modular Robotic Cubes: Toward Replicating Morphogenetic Movements of the Embryo

In this paper we present a new type of simple, pneumatically actuated, soft modular robotic system that can reproduce fundamental cell behaviors observed during morphogenesis; the initial shaping stage of the living embryo. The fabrication method uses soft lithography for producing composite elastomeric hollow cubes and permanent magnets as passive docking mechanism. Actuation is achieved by controlling the internal pressurization of cubes with external micro air pumps. Our experiments show how simple soft robotic modules can serve to reproduce to great extend the overall mechanics of collective cell migration, delamination, invagination, involution, epiboly and even simple forms of self-reconfiguration. Instead of relying in complex rigid onboard docking hardware, we exploit the coordinated inflation/deflation of modules as a simple mechanism to detach/attach modules and even rearrange the spatial position of components. Our results suggest new avenues for producing inexpensive, yet functioning, synthetic morphogenetic systems and provide new tangible models of cell behavior.


Hydrodynamic capacitance
The hydrodynamic capacitance in a fluidic system is a parameter that describes the change in the volume of a fluid, enclosed in a channel or chamber, in front of a change in pressure. This phenomenon occurs due to the compressibility of the fluid or the elasticity of the chamber.

Compressibility of a fluid
The compressibility of a fluid is measured by: , where is the initial volume of the fluid (equals to the volume of the chamber), the change in the volume of the fluid, and the change in the pressure of the fluid. The relation has negative sign because an increment on the pressure applied to a compressible fluid results in a reduction in its volume. The compressibility of the air is = 1 . Thus, the flow rate entering a chamber of volume , because of fluid compression ∆ , is: , and the hydrodynamic capacitance of the fluid is therefore: = .

Dilatability of a channel or a chamber
The dilatability of a channel or a chamber is defined by: , and the hydrodynamic capacitance of the chamber is therefore: Notice that the volume of the cube, the strain capacity of the silicone and the compressibility of the air defines the hydrodynamic capacitance of the system. For example, the larger the strain capacity, the larger the hydrodynamic capacitance of the cube. The lower the compressibility of the fluid, the lower the capacitance of the system.
Hydrodynamic capacity of the system Our fluidic system presents both characteristics: compressible fluid (air) and elastic chamber. Hence, following the analogy between electronics and fluidic systems, the total hydrodynamic capacitance of the system is:

= +
, and the final dynamic response of the flow rate inflating the module is:

Dynamic response of the pressure inside a module
To understand the dynamics of the pressure developed inside a module, , we assume that its initial pressure is the atmospheric pressure and that the hose is not elastic. To continue, corresponds to the pressure generated at the air pump minus the pressure drop along the hose, as described by Eq S1 rearranged as follows: However, as the flow varies as the module fills, we must replace Eq S2 in Eq S3 as follows: , which implies that: Eq S4 constitutes a first order system whose solution for is given by the following expression: Eq S5 shows that the time constant of the system is 1 = ; the larger the time constant, the slower the step response of the system. Also, we know that geometry of the hose affects and that strain capacity of the silicone affects .
Impact of the characteristics of the air pump is a characteristic of the pump and defines the maximum pressure that can be built inside a module.
Nonetheless, maximum flow rate of the pump is another important characteristic since it determines the speed at which the maximum pressure is built. In the formulation of Eq S5 we assumed that the pump can provide any required flow to build up instantaneously, which is true as long as the maximum flow required by the module is below the maximum flow rate of the pump. The maximum flow rate required by the module is defined by the following equation: = , assuming that is a gauge pressure; i.e., absolute pressure minus atmospheric pressure.

Dynamic response of the volumetric expansion of a module
When determining the hydrodynamic capacitance of the chamber, we just considered ℎ , which can be obtained with the initial and final volume of a module. Nonetheless, the dynamics of the volumetric expansion is also of interest, for example, to analyze the transient response of a group of connected modules undergoing actuation at different phases. To obtain a model, we first measured the volumetric expansion of a module that resulted when slowly increasing the pressure , as shown in Fig. 3a. For each measurement point, volumetric expansion was measured after the system stabilized.
Although we measured in this experiment (not ) at the gauge location (See S6 Fig), the volumetric expansion was measured once the system reached its stationary state, therefore = . To continue, Fig. 3a also shows that the volumetric response has an asymptote at the pressure: the maximum resistance of a module, . With this in mind, we fitted a model to the measured volumetric response, as follows: , where ∆ is the volumetric expansion of a module that triggers an accelerated volumetric expansion.
As seen in Fig. 3a, the maximum pressure our modules can sustain is about ≈ 15.6 . Fig. 3a  ), Eq S6 can be expressed in terms of volume, as follows: where 0 is the initial volume of the cube, and = ∆ 0 100 + 0 is the volume of the cube that triggers an accelerated volumetric expansion.
We can also find the volumetric expansion of a module as a function of time by replacing Eq. S5 into Eq.
S6, as follows: which can also be expressed in terms of volume, as follows: On a second experiment we measured the transient volumetric response of a module to an impulse. This was implemented by applying a large pneumatic pressure = 138 kPa over a short period of time (nearly 50ms). Figure  A breakout expansion test was performed with a module inside the beaker filled with water.
Volumetric expansion was also measured by observing water displacement as the compressor injected air through the hose. As a result, the modules expanded up to 1460% (125cc) before bubbles appeared indicating failure.
In order to determine the speed of reaction of the system, the step response of the volumetric expansion was registered. The magnitude of the pressure step was 137.9kPa. The instantaneous volume was estimated using video frames. The resulting measured expansion speed was rather high since the module reached its final volume after 200ms, as shown in Fig 3a.