Bioinspired soft bendable peristaltic pump exploiting ballooning for high volume throughput

—Interest in bioinspired peristaltic pumps has grown in popularity among the scientiﬁc community in the last decade thanks to their extreme ﬂexibility and their intrinsic compliance. In this paper, we propose a soft peristaltic pump exploiting ballooning. Our aim is to promote and propel forward the ballooned region by controlling the air pressure between the balloon and an external ﬂexible containment tube. Thanks to this mechanism, it is possible to achieve a peristaltic pumping motion with a simple design and using only one control signal. In this paper, we describe the implementation of the pump and the inlet-pump-outlet system, provide an analytical model to predict the pump performance, and experimentally test the device. Finally, the proposed pump is directly compared with the state-of-the-art. We show that it is possible to achieve high ﬂow rates, up to 4 . 5 mLs , with only a single control signal and relying on a much simpler design, paving the way for more ﬂexible and easy to manufacture peristaltic pumps.


I. INTRODUCTION
Robotics has become integral to everyday life in modern society, but rigid robots show strong limitations in compliance, safety, and flexibility. This is where soft robotics excels, using stretchable and flexible materials as structural elements and actuators [1], [2], [3]. Considering the subdomain of peristaltic pumps, rigid pumps achieve the characteristic peristaltic motion by temporally crimping the tube with rollers [4], potentially cutting and damaging the tube if the medium contains solid particles. Conversely, soft pumps reproduce the peristaltic motion observed in biological systems [5] by understanding and replicating motion primitives observed in nature [6], [7], [8], achieving high compliance and good flexibility. The bending capability of such devices makes them suitable for performing actuation in narrow and tight spaces with obstacles, such as medical applications [9].
The actuation technologies that have been exploited so far for peristaltic actuation are dielectric active elastomers (DAEs) [10], [11], [12], magnetic actuators (MAs), shape memory alloys (SMAs) [13], [14] and pneumatic artificial muscles (PAMs) [15], [16], [17]. Among these, DAEs and MAs are able to achieve high frequencies, up to 10 Hz, but the limited stroke is unable to provide high flow rates. Instead, PAMs showcase the best flow rate, around 10 mL s , thanks to a modular architecture bioinspired from the small intestine [18], [19], [20]. Although powerful, this solution shows extreme complexity in fabrication, control, and scalability. The fabrication of each actuator is performed through silicone casting which requires considerable time for manual fabrication. In addition, due to the modular design, each 1 The Bio-Inspired Robotics Lab, Department of Engineering, University of Cambridge, UK. * fi224@cam.ac.uk.com module needs to be connected to the air supply and controlled independently. Overall, state-of-the-art pumps show either a low flow rate or extreme complexity and limited flexibility due to the wiring.
In this work, we propose a non-modular design exploiting and controlling the buckling of an elastomeric balloon, known as ballooning. Ballooning is the phenomenon by which balloons, when inflated, rapidly create a localized completely expanded region, with little deformation of the rest of the structure [21]. Theoretically, the position of the ballooned region along the balloon can be easily changed since all possible positions are energetically equivalent [21], [22]. Our aim is to promote ballooning and then propel the inflated region forward, after which reversing the buckling at the end of the pump to create a peristaltic pumping motion that can then be cycled. However, controlling buckling is a complex and problematic task, since it is an unstable phenomenon very sensitive to any change in geometry or load [23], [24], [25].
We propose using an external flexible tube to constrain the balloon. If the inner diameter of the tube is smaller than the diameter of the inflated balloon, the ballooning would seal the tube and create two distinct regions: one proximal, between the inlet and the inflated region, and the other distal, between inflated region and outlet. By constraining it in this way, we can then use a single control pressure signal on the proximal region to achieve the peristaltic motion, allowing on/off control to trigger pumping. Finally, by developing a mathematical model of the inlet-pump-outlet system, we aim to give an additional tool for designing soft pumps that exploit ballooning.
In the remainder of the paper, we present the pump design and fabrication in Section II-A, followed by the overall setup in Section II-B, the mathematical model in Section II-C, and the experimental design and results in Sections II-D and III.

A. Pump design and fabrication
The proposed pump design consists of two main components: the balloon and the external tube. The balloon is kept in position inside the tube by two supports that also connect the balloon to the inlet and outlet tubes. The left support also provides the connection to the air supply from the pneumatic regulator so that the pressure between balloon and tube can be regulated and controlled (see Fig. 1).
For the fabrication, we used a 2.2 · 10 −3 cm thick natural latex balloon (Pengxiaomei, Modelling Balloons) [26], [27] and a PET-PVC external tube (RS Components, 368-0233) [28]. The external tube's length is 45 cm to match the partially inflated balloon's length, and its internal diameter is 0.126 cm to ensure a tight junction with the balloon. The inlet and outlet supports have been 3D printed with ABS to fit the ends of the pump and host the needed connections. All the components are assembled together by hand.
In order to achieve spontaneous ballooning, the device needs to be placed at a lower level than the water source connected to the inlet, so that the fluid pressure inside the balloon can trigger the buckling, and the balloon presents itself with an inflated region. Experimentally has been observed that placing the device at a distance h = 160 cm below the water source produces a spontaneously inflated region of length l p = 20 cm. This phenomenon is more likely to happen in the proximal section because of balloons' imperfections.
The control of the pump is open-loop and the control signal used to achieve peristaltic motion is a square pressure wave between balloon and tube, supplied by the pneumatic regulator. The 3 tunable parameters are the maximum pressure P and the activation and rest time, t on and t of f respectively (Fig. 2).
The pumping cycle can be divided into four parts: buckling, recoil, inverse buckling, and propulsion (see Fig. 3). Buckling and recoil happen when the pressure between the balloon and the external tube is at the atmospheric level: buckling happens when the pneumatic regulator is switched  off, due to the differential pressure between inside and outside the balloon, whereas recoil is the passive flow from inlet to outlet, due to a lack of pressure in the outlet following the ballooning. Conversely, propulsion and inverse buckling happen when the pneumatic regulator is switched on. The buckled balloon divides the external tube is two regions, one proximal and one distal: only the proximal is pressurized since the regulator is connected on the inlet side, resulting in a force propelling the inflated region forward.

B. Setup
Once the pump is assembled, it is connected to the inlet and the outlet tubes, which connect the pump with both source and target. The left support is also connected to the pneumatic regulator, which uses an air compressor operating at 10 bar as a source. The pneumatic regulator is controlled with Arduino UNO to achieve the square pressure wave. The pump is kept at a distance h below the fluid source and the target reservoir. Fig. 4 shows the experimental set-up and a corresponding schematic.

C. Mathematical model
In this section, a mathematical model of the overall system is proposed in order to predict the performance as a function of the square pressure wave's parameters t on and t of f . The following demonstration is divided into the four phases of the pumping cycle: buckling, recoil, inverse buckling, and  propulsion. For ease of discussion we will discuss the process in this order, whereas in the real system, propulsion happens before inverse buckling. Since the buckling is a much faster phenomenon than recoil or propulsion of the inflated region, both the buckling and the inverse buckling will be considered instantaneous. All pressure values are referred to as gauge pressure.
1) Buckling: When the regulator switches off, the fluid's pressure ρgh (where ρ is the density of the inner fluid and g is the gravitational acceleration) causes ballooning, creating an inflated region that is filled with water from both outlet and inlet (see Fig. 5 a and b). Assuming that this phase happens instantly, it is supposed that the volume of water introduced in the pump V p = l p π(R 2 p − r 2 p ) is provided by inlet and outlet depending on their distance from the middle point of the inflated region: where O o1 and O i1 are the volume of fluid provided by the outlet and the inlet, respectively, L i1 is the length of the inlet tube plus the distance from the start of the pump until the center of the inflated region and L o1 is the length of the outlet tube plus the remaining length of the pump, at this specific phase. The index 1 is used to indicate the first phase: buckling. Then, it is possible to compute the height , which is the starting point of the recoil.
2) Recoil: The recoil describes how the fluid level of the outlet raises to match the inlet. During this phase the entire system can be schematize as a fluid reservoir with an orifice at a distance d below the level of the fluid (see Fig. 5 c and d). The solution of such a model is known as Torricelli's law, but in our case the distance d(t) is a function of the velocity of the fluid at the orifice, v(t), leading to the following set of equations: Substituting in and integrating, you can get: where V (t) is the anti-derivative of v(t). Finally, by deriving with respect to t we obtain the fluid's acceleration asv(t) = −g, hence both v(t) and d(t) can be rewritten as: Finally, we can compute the volume of fluid that is still needed to completely fill the outlet tube at the end of the recoil as M = πr 2 o d(t of f ).

3) Inverse buckling:
This phase is the only one in which we have a net positive flow rate going into the receiving container (see Fig. 5 e and f). The total volume of water ejected is: where the first term is due to the inverse buckling and the second one is due to the rebound of the balloon itself after. Following the same assumption of the buckling phase and correcting the outlet volume by the volume M obtained during the recoil phase, it is possible to compute the volume of water ejected by inverse buckling as follows: where V p is the volume of fluid inside the balloon before buckling, O o3 and O i3 are the volume of fluid ejected in the outlet's and inlet's reservoirs, respectively, L i3 is the length of the inlet tube plus the distance from the start of the pump until the center of the inflated region and L o3 is the length of the outlet tube plus the remaining length of the pump, at this specific phase. The index 3 is used to indicate the third phase: inverse buckling. Note that the inverse buckling happens distally, whereas the buckling happens proximally.
The second contribution is due to the rebound of the balloon. When an instantaneous change in volume happens (i.e. inverse buckling), the volume between the balloon and the tube increases suddenly, thus decreasing the pressure. Simultaneously, the air compressor tries to bring the pressure back up while the balloon would start inflating due to the fluid pressure on the inside. Those two phenomena cause a rebound of the balloon and more fluid is ejected as a consequence. Given the complexity of this problem, an analytical formulation is not provided, and the magnitude of the rebound is approximated as a second-order decaying phenomenon with respect to time: A(t) = (at 2 + bt + c). The rebound's contribution O reb can then be computed as follows: The parameters a, b and c are selected using the Nelder-Mead simplex optimization algorithm [29] on the experimental data: a = −3.9941 × 10 − 10 mL s 3 , b = −4.9844 × 10 − 7 mL s 2 and c = 6.6 × 10 − 3 mL s . 4) Propulsion: Upon activation of the regulator, the airflow introduced pushes the inflated region of the balloon from the proximal region to the distal one, where the inverse buckling happens. In the experiments the needed time for this process has been measured to be τ = 500 ms for a 0.4 bar air pressure. Taking into consideration this time delay, the average flow rate of the pump F can be predicted as follows: where u(t) is the unit step function.

D. Experiments
When testing the pump, the main outcome considered is the average flow rate: the pump is activated for five cycles and the total volume of water is then divided by the time needed to complete the cycles. All experiments are run using water and every experiment is repeated 6 times with different balloons. Since the only control signal is a square pressure wave, the only 3 controllable parameters are P , t on , and t of f . The parameters are initialized at 0.4 bar, 1650 ms and 7000 ms, respectively, and are optimized individually: first t on , then t of f and lastly P . Next, the pump's performance in 5 configurations of increasing bending radius, up to 360 deg, is analyzed. Finally, the pump is also tested using coffee powder to simulate solid particles inside the water.

III. RESULTS
First, keeping P = 0.4 bar and t of f = 7000 ms, the pump has been tested and its performance has been evaluated as a function of t on ( Fig. 6 a). Next, fixing t on = 1650 ms, which corresponded to the maximum flow, the performance has been evaluated as a function of t of f (see Fig. 6 b). It can be noticed that the best performance is obtained for t on = 1650 ms and t of f = 3000 ms, but the pump is able to achieve a good performance up to a frequency of 1 Hz, with t on = 500 ms and t of f = 500 ms. If t on is not enough to trigger the inverse buckling, there is a drop in performance due to the inflated region of the balloon not contributing to the ejected volume. It is also clear that the rest time needs to be sufficient for the fluid recoil to happen, otherwise, the majority of the ejected volume is wasted in the outlet tube and does not contribute to the pump's flow rate. Moreover, the proposed mathematical model is shown to enable accurate prediction of the pumps performance.
Next, keeping constant the optimized parameters t on = 1650 ms and t of f = 3000 ms, the effect of the pressure's amplitude is investigated (see Fig. 7). The bar chart shows that both low and high pressures result in a substantial Each tested value has been sampled 6 times using 6 different balloons. decrease in performance. Analytically, increasing the operating pressure results in a lower τ and lower magnitude of the rebound A(t). With a pressure of 0.2 bar, the force exerted on the inflated region is not sufficient to propel it forward and to promote the inverse buckling, lowering the total volume of fluid ejected per cycle. On the other hand, high values of pressure (0.6 bar and 0.8 bar) are achieved by increasing the pneumatic regulator's airflow rate, which decreases significantly the magnitude of A(t) (see Eq. 5).
To test the flexibility of the proposed pump design, the best operational parameters P = 0.4 bar, t on = 1650 ms and t of f = 3000 ms are tested while bending the pump in different configurations. Fig. 8 shows the relative flow rate with respect to the unbent configuration and how the different orientations are achieved. The results show that the effect of bending is limited at most to 11%, thus the pump could be used in clustered and dynamic environments and could be bent around obstacles without extreme losses, making it suited for tasks in narrow spaces with obstacles to avoid, such as surgical theaters and healthcare facilities [31]. Finally, to investigate the effect of solid particles in the medium, the device is tested pumping a slurry obtained by mixing 150 g of coffee powder and 4 L of water, with the same optimal parameters. We run the experiments with both water and the mixture, and while pumping the latter the flow rate is only 0.6 ± 0.2% lower, showing that the presence of the powder does not affect at all the performance.
When compared with other state-of-the-art biomimetic peristaltic soft pumps (see Table I), this proposed solution shows good performance, in particular for the flow rate, which is second only to modular PAMs. In addition, the pump showcases an extremely simple design, fabrication process, and control strategy, with only one control signal required to achieve its high flow rate.

IV. CONCLUSION
This work proposes an innovative strategy to achieve bioinspired peristaltic soft pumping: controlling the spontaneous buckling of a balloon filled with fluid. The pump is composed only of the balloon containing the medium and an external tube, resulting in a simple design and an easy fabrication process. Our pump needs to be placed at a given distance below the level of the source and target reservoirs to trigger the instability, but this constraint could be avoided by promoting the ballooning with a vacuum between the balloon and the external tube. Note that this could be achieved using the same pneumatic line to impose vacuum instead of atmospheric pressure during buckling and recoil, thus not needing a second one. The inflated region is moved using a single pneumatic regulator connected to the proximal end. A mathematical model is introduced to predict the performance of the pump, considering the overall testing bench. The effects of amplitude, activation time, and  [11] Dielectric active elastomer 1.5 5 1 − − Lotz [12] Dielectric active elastomer 0.1 3.5 1 1.8 × 10 −4 10 Fuhrer et al. [30] Magnetic actuation 0.95 24 4 1.3 4 Miki et al. [13] Shape memory alloy 1.8 3.5 2 3 − 6 0.33 Sun et al. [14] Shape memory alloy 0.9 8 3 0.013 0.1 − 2 Suzuki et al. [18] Pneumatic artificial muscle 3 23.5 5 33 1.2 − 2 Dirven et al. [16] Pneumatic artificial muscle 1.8 20 12 5 − 10 − Esser et al. [9] Pneumatic artificial muscle rest time on the flow rate have been analyzed to provide a complete characterization. When compared with state-ofthe-art soft peristaltic pumps, our approach is shown to have a high flow rate: much greater than the other single input systems and second only to complex modular designs. Our device maintains high volume throughput until 1 Hz before strongly decreasing its performance, due to failed propulsion and recoil's time scale. Moreover, our design is proven to be much simpler than the current state-of-the-art, given the need for only one control signal, the faster and easier fabrication process, and the scalability of the device. Unlike the other pneumatic solutions in literature, the proposed pump is not modular, thus the length of the pump itself does not affect the fabrication nor the number of control signals needed to operate it. Finally, the extreme versatility of the device is tested by actuating it in various bent configurations and using different media, and the performances are only slightly affected. In summary, utilizing buckling provides many advantages to the pumping mechanism, enabling many novel functionalities and state-of-the-art performance. Future work could include the parallel implementation of different balloons pumping different media and the systematic introduction of weak points to better control and stabilize the buckling.
ACKNOWLEDGMENT This work was supported by the SMART project, European Union's Horizon 2020 research and innovation under the Marie Sklodowska-Curie (grant agreement ID 860108).