Microfluidic Circuit Dynamics and Control for Caterpillar-Inspired Locomotion in a Soft Robot
Helene Nguewou-Hyousse1, Giulia Franchi2, and Derek A. Paley3
Abstract— Because caterpillars can locomote through com- The motivation behind this work is the design and plex terrain, caterpillar-inspired soft robots promise safe and closed-loop control of a novel locomotion mechanism in reliable access for search, rescue, and exploration. Microfluidic a caterpillar-inspired soft robot. We seek to design a 3D- components analogous to electronic components like resistors and capacitors further promise the principled design and printed caterpillar robot with internal microfluidic circuitry fabrication of autonomous soft robots powered by microfluidic that drives leg actuation.When an actual caterpillar moves, circuits. This paper presents a model for caterpillar locomotion each left-right pair of legs is lifted simultaneously and goes using an oscillator network approach, in which the periodic through a stance phase and a swing phase [5], [6]. In addi- motion of each leg is described by an oscillator, and the tion, a wave is repeatedly generated at the terminal proleg collection of all legs forms a network. We use tools from graph theory to model (1) a first-order system consisting of a and travels to the front leg, creating a periodic behavior network of RC oscillators connected to an astable multivibrator that can be modeled by an oscillator network with a cyclic circuit serving as a Central Pattern Generator (CPG); and (2) a topology. second-order system consisting of a network of RLC oscillators Electrical circuits may be used to represent microfluidic with a deCentralized Pattern Generator (dCPG) as reference circuits using Equivalent Circuit (EC) Theory [7]. Although control. For the RC network, we analyze the rate of convergence and the gait number, a metric that characterizes the locomotion not all electrical components have a microfluidic analog gait. For the RLC network, we design feedback laws to stabilize (e.g., there are no microfluidic inductors due to low inertial consensus and traveling wave solutions. Numerical modeling effects at the millimeter scale), EC supports the design of and preliminary experimental results show promise for the microfluidic circuits and robots. For example, the Octobot is design and control of locomotion circuitry in a caterpillar- an octopus-inspired autonomous soft robot with microfluidic inspired soft robot. logic fabricated using soft lithography [1]. However, recent trends in microfluidics have demonstrated the viability of 3D- I.INTRODUCTION printed circuit components [8]; using this technique to print Robots made from soft materials are more flexible and a microfluidic soft robot would reduce the need for circuit may be more capable of adapting to complex environments interconnections and may simplify design and fabrication. than rigid robots. As such, their applications range from An astable multivibrator [1], [9], for example, is an oscillat- search and rescue to exploration and medicine [1], [2]. Soft ing circuit that can be implemented with either electric or robots inspired by organisms without a skeleton may also microfluidic components. be easier to design and fabricate than vertebrate-inspired This paper considers a coordinated network of fluidic robots, because of the level of abstraction offered by the oscillators generating and controlling periodic leg actua- simplicity of their anatomy [3]. A caterpillar is an example tion. Previous work on modeling caterpillar-inspired loco- of a soft-bodied animal that can navigate irregular and motion in a soft body applied inverse-dynamics to a planar vertical terrains [4]. Caterpillar locomotion exhibits (i) the extensible-link model [10], or lumped-mass dynamic model synchronous motion of left-right pairs of legs, and (ii) a wave coupled with a control law for gait optimization [2], [4]. propagation pattern traveling from tail to head [4]–[6]. Thus, Autonomous decentralized control is a solution to generating a caterpillar-inspired soft-robotic system should be capable and controlling local and adaptive locomotion in soft robots of collective behavior such as synchronization and wave [11], [12]. In particular, Umedachi and Trimmer imple- propagation. Furthermore, choosing appropriate fabrication mented an autonomous decentralized control to extend and material [3] and structure geometry should guarantee its contract segments of a 3D printed modular soft robot; the flexibility. results show that implementing modular deformity resulted in a phase gradient that induced locomotion [11]. Prior work in the area of collective motion of networked *This work is supported by ARO Grant No. W911NF1610244 1Helene Nguewou-Hyousse is a graduate student in the Department of oscillators used a spatial approach to look at the phases Electrical and Computer Engineering, University of Maryland, College Park, of each oscillator [13], [14]. Here we investigate feedback MD 20742, USA [email protected] control of an oscillator network to generate cyclic actuation 2Giulia Franchi is Assistant Professor of Computer Science in the Department of Mathematics and Computer Science, Salisbury University, of many individual legs. We study both first- and second- Salisbury, MD 21801, USA [email protected] order dynamical models. For the first-order model, which 3Derek A. Paley is the Willis H. Young Jr. Professor of Aerospace consists of a multivibrator circuit driving a set of RC oscil- Engineering Education in the Department of Aerospace Engineering and the Institute for Systems Research, University of Maryland, College Park, lators in series, we introduce a gait number to characterize MD 20742, USA [email protected] the locomotion pattern. For the second-order model, which consists of a network of RLC circuits in a cyclic topology, the following linear state-space representation: we modify the network to include a deCentralized Pattern d v 0 1 v 0 = + V (t). Generator (dCPG) as the decentralized control. For example, dt a − 1 − r a 1 in consensus control of multiple oscillators, the dCPG sets lc l lc 2 the collective phase, but the phase lags if the network itself The circuit oscillates if it is underdamped, i.e., when r c < is lagging. 4l. In the absence of an inductor, i.e., in a microfluidic circuit, The contributions of this paper are (1) the design of a first- we have a resistor-capacitor (RC) circuit in series, which is order RC oscillator network model for generating caterpillar- governed by the first-order equation v 1 inspired locomotion in a microfluidic circuit, and the charac- cv˙ + = V (t). terization of the resulting locomotion gait; (2) the design of r r a second-order RLC oscillator network model, and closed- The time constant of τ1 = rc represents the time required loop feedback control laws for consensus and traveling wave for the voltage at the capacitor to drop to v(0)/e. solutions; and (3) the inclusion of a deCentralized Pattern B. Algebraic Graph Theory Generator to regulate the collective phase of the consensus Graph theory is used to represent the topology of the solution in the RLC network. This work demonstrates coop- oscillator network in which each leg is represented as a node. erative behavior of a network of oscillator circuits compatible The Laplacian matrix L defines the connections between the with microfluidics; it also proposes a model for the closed- nodes of the network. Given the undirected graph G = (V, E) loop control design of an RLC network to produce stable with vertex set V of cardinality N and edge set E, we write cyclic motion. These steps provide a theoretical approach to the Laplacian matrix as L = D − A. The adjacency matrix the study of legged locomotion in an autonomous soft robot A ∈ N×N is symmetric, with entries a = a given by using microfluidics; we also include preliminary results from R ij ji ( our ongoing experimental testbed. 1, if i 6= j and (i, j) ∈ E aij = The paper is organized as follows. Section II presents 0, otherwise. an overview of RLC oscillators, algebraic graph theory, N×N th first- and second-order consensus dynamics, and microfluidic The degree matrix D ∈ R has the ij entry circuit theory. Section III-A provides a mathematical frame- (PN j=1 aij, if i = j dij = work for the RC network model and presents an analysis 0, otherwise. of its convergence properties, including a parametrization of the locomotion gait. Section III-B contains a mathemat- The Laplacian of an undirected graph is a symmetric, pos- ical framework for the RLC network, including open-loop itive semi-definite matrix with zero row sums. For example, stability analysis, closed-loop control design for consensus the Laplacian matrix of an undirected chain graph with N and traveling wave solutions, and the effect of the dCPG. edges is Section IV presents preliminary experimental results from ongoing work with the RC microfluidic network. Section V 1 −1 0 ··· 0 summarizes the paper and our future work. −1 2 −1 0 ··· 0 0 −1 2 −1 0 ··· 0 ...... II.BACKGROUND ...... N×N L1 = ∈ R , ...... The proposed work models each leg of a caterpillar as . . . . . an RLC (or RC) circuit. The network of N caterpillar legs . . . .. −1 2 −1 is connected in either a circulant or a chain topology. In 0 0 ··· 0 −1 1 the microfluidic testbed, pressure is analogous to voltage, (1) whereas volumetric flow rate is analogous to current. In a whereas the Laplacian matrix of an undirected cyclic graph pneumatic setting, each microfluidic leg will be actuated by with degree 2 is internal pressure; we use Equivalent Circuit theory to design these pressure inputs using electrical circuits. This section 2 −1 0 · · · −1 first reviews RLC and RC circuits, followed by a background −1 2 −1 0 ··· 0 overview of graph theory, consensus dynamics, and the field 0 −1 2 −1 0 ··· 0 of microfluidics. ...... ...... N×N L2 = ∈ R . ...... A. RLC and RC Circuit Theory . . . . . . . Kirchoff’s loop equation for a series RLC circuit with . .. −1 2 −1 resistance r, inductance l, capacitance c, current i, and input −1 0 ··· 0 −1 2 voltage V (t) is vr + vc + vl = V (t), where vl = ldi/dt, (2) 1 R t vr = ri and vc = c −∞ i(τ)d(τ) [15]. Defining state Note that L2 is a circulant matrix, i.e, each row vector is a variables v = vc and a =v ˙c, the system dynamics have cyclic permutation of the preceding row vector. C. Consensus Theory Depending on the network topology, it is often possible to achieve collective behavior via consensus. Consensus refers to a group of agents achieving a common decision or behavior over time through a local distributed protocol [16]–[18]. Consider the linear consensus scheme [19] v˙ = −Lv,
T N where v = [v1, ··· , vN ] ∈ R and L is a graph Laplacian. Previous work [19], [20] showed that this system eventually achieves first-order consensus, i.e., vj = vk for all pairs j and k, if the associated communication graph G has a spanning Fig. 1: Schematic of an astable multivibrator connected to tree. G has a directed spanning tree if and only if L has a an RC network with N = 4 simple zero eigenvalue (with associated eigenvector 1N ) and all other eigenvalues have positive real parts. Furthermore, −Lt T PN e → 1N p and vi(t) → i=1(pivi(0)) as t → ∞, T III. DYNAMICAL MODELS where p = [p1, ··· , pN ] ≥ 0 is a left eigenvector of L associated with the simple zero eigenvalue and PN p = 1. i=1 i This section contains the mathematical framework govern- Consider N coupled harmonic oscillators with local inter- ing the dynamics of two oscillator networks. Section III-A action of the form [16] describes an RC network driven by a multivibrator oscillator, d v 0 I v and Section III-B describes an RLC network with control = N N , dt a −αIN −L a inputs at each node. The RLC network incorporates a dCPG as reference input to regulate the collective phase. where 0 denotes the N × N zero matrix , I denotes the N × N identity matrix, and L is a Laplacian matrix associ- ated with directed graph G. There exists right eigenvector A. First-Order System: RC Network N 1N satisfying L1N = 0 and left eigenvector p ∈ R T T satisfying p ≥ 0, p L = 0, and p 1N = 1. The Motivated by the existence of microfluidic transistors, N agents achieve second-order synchronization if G has capacitors, and resistances, this design includes an astable multivibrator connected to N resistor-capacitor (RC) circuits a directed√ spanning tree√ [16]. More√ specifically, vi(t) → cos αtpT v(0) + (1/ α) sin αtpT a(0) and a (t) → as shown in Fig. 1. To create a phase delay between each RC √ √ √ i − α sin αtpT v(0) + cos αtpT a(0) as t → ∞. circuit, the network has a chain topology in which each node is connected to its two nearest neighbors, with the exception D. Microfluidic Circuit Theory of the first element, which is connected to the multivibrator Microfluidics refers to the precise manipulation of fluidic and one neighbor, and the last element, which is connected networks at the submillimiter level [21], [22]. Microfluidic to only one neighbor. The multivibrator is a transistor circuit components include diodes, capacitors, transistors, and hy- whose output is a square wave that oscillates continuously draulic resistances. Microfluidic circuits can be analyzed between high and low values [9]. When one transistor (Q1) using computational fluid dynamics (CFD) analysis, although is switched on, the other (Q2) is switched off. At the same as the number of circuit components to be modeled increases, time, capacitor C1 discharges while C2 charges. the CFD analysis becomes complex. Electric circuit theory Let L1 be the Laplacian matrix associated with a chain is an alternative technique that uses the analogy between topology, and vi be the voltage associated with capacitor i. T fluidic and electronic components to describe the dynamics Let v = (v1, ··· , vN ) and v0 be the voltage at the transistor of a microfluidic circuit using electrical equations [22], Q2, i.e., the output of the multivibrator. During the high [7]. Pressure, volumetric flow rate, and hydraulic resistance (or low) output of the multivibrator, v0 is constant, and the are equivalent to voltage, current, and electric resistance, dynamics of this network system are respectively. By making the assumption that the fluidic flow is laminar, viscous, and incompressible, the use of electric 1 1 v˙ = − L v + (v − v ), circuit theory greatly simplifies the analysis of the flow rc 1 rc 0 1 dynamics in microfluidic circuits. Microfluidic circuit theory applied to robotics is relevant because it promises to enable where L1 is the Laplacian matrix defined in (1). Using T the design and fabrication of autonomous microfluidic soft augmented state v˜ = (v0, ··· , vN ) and Laplacian L˜1 ∈ robots; moreover, using 3D-printing techniques to fabricate RN+1×N+1, the dynamics become the microfluidic components and integrate with a soft body may allow the fabrication of an autonomous fully integrated 1 v˜˙ = − L˜ v˜, (3) soft robot [21]. rc 1 6
5
4
3
2
Voltage (V) Fig. 3: Network of N RLC circuits connected through an
v 0 edge resistance 1 v 1 v 2 0 v 3 the lack of microfluidic inductors, it has several attractive v 4 features for control design of networked oscillators. In par- -1 0 0.5 1 1.5 2 2.5 3 3.5 ticular, the RLC model illustrates how feedback at the circuit time (s) level might be used to achieve caterpillar-inspired locomotion including consensus and traveling wave solutions. Fig. 2: Evolution of N = 4 RC oscillators with CPG v0 Consider an undirected graph with a circulant topology of N nodes. This graph describes the network of identical RLC oscillators coupled through an edge resistance depicted where in Fig. 3. Suppose the kth RLC series circuit has resistance 0 ··· 0 rk = r, inductance lk = l, capacitance ck = c, input −1 2 −1 0 ··· 0 voltage Vk(t) and edge resistance rjk =r ¯. In addition, T 0 −1 2 −1 0 ··· 0 the state of oscillator k is [vk ak] , k = 1,...,N. Let T T 0 0 −1 2 −1 0 ··· 0 v = [v1, . . . , vN ] and a = [a1, . . . , aN ] , as before, so ˜ ...... that the dynamics of the oscillator are L1 = ...... . ...... d v v 0 ...... = M + V (t), (5) a a 1 I . . . dt lc . . .. −1 2 −1 where 0 0 0 ··· 0 −1 1 0 I Equation (3) essentially includes the astable multivibra- M = 1 r r 1 , − I − L2 − I − L2 tor as a Central Pattern Generator that drives the network lc lcr¯ l cr¯ T dynamics. Fig. 2 shows the results of a Matlab simulation V (t) = [V1(t), ··· ,VN (t)] is the collection of voltage inputs, one per node, and L is the circulant Laplacian matrix with v0 = 5.2V , r = 1 kΩ, R = 10 kΩ, and C1 = C2 = 2 c = 0.1 mF . Note that, for the parameters chosen, the RC defined in (2). T oscillators do not reach consensus, but instead show a phase Let wr = [xr, yr] be a right eigenvector of M associated lag in the rising voltage of consecutive oscillators. with eigenvalue λ, such that Mwr = λwr. We have We formalize this idea as follows. The period of an astable yr = λxr multivibrator is τ = ln 2(R C + R C ) and τ = rc is the 0 1 1 2 2 1 −1 r r 1 time constant of the RC circuit. Define the gait number I − L x − λ I + L x = λ2x , lc lcr¯ 2 r l cr¯ 2 r r τ η = 0 (4) which implies Nτ1 2 r 1 to characterize the caterpillar-inspired gait with N legs. λ + λ l + lc L2xr = −cr¯ r xr. (6) Let C1 = C2 = c and R1 = R2 = R. Then, η = l + λ 2 ln (2)R/(Nr). Observations from simulations at various η | {z } values show that when η < 1, the contribution of the front ,µ legs to the caterpillar locomotion is negligible. On the other Let µ be an eigenvalue of L2 associated with the vector hand, when η > 1, all legs contribute to the locomotion. χr = xr. λ is a solution of η rcr¯ r¯ r Furthermore, as increases, the voltage outputs of legs crλ¯ 2 + + µ λ + + µ = 0. 1 through N increases to v0(t), and the phase difference l l l between the leg nodes decreases to 0. For the eigenvalue µi of L2 associated with eigenvector χri, we can therefore define an eigenvalue of M with associated B. Second-Order System: RLC Network T right eigenvector wri = [χri, λi±χri] as follows: Next, consider a second-order model that represents each q rcr¯ rcr¯ 2 r¯ r leg of the caterpillar as an RLC circuit. Although this model −( l + µi) ± ( l + µi) − 4cr¯( l + l µi) λ = . is currently unattainable in microfluidic circuitry, because of i± 2cr¯ In a similar way, if we let w = [x , y ]T be the left z z z 1.5 eigenvectors of M associated with the eigenvalue λ such that wT M = λwT , we have z z 1 T −1 r T yz I − L2 = λxz lc lcr¯ 0.5 r 1 x T − y T I + L = λy T , 0 z z l cr¯ 2 z Voltage (V)
-0.5 v which implies 1 v 2 r 1 v 2 3 λ + λ + -1 T l lc T v yz L2 = cr¯ r yz , (7) 4 + λ v l 5 -1.5 Note that (7) is similar to (6). Let χz = yz be 0 0.5 1 1.5 2 2.5 3 3.5 the left eigenvector of L2 associated with eigenvalue time (s) µ. Thus, the corresponding left eigenvector of M is T 1.5 wzi=[1/λ(−1/(lc)I − r/(lcr¯)L2)χzi χzi] . We have the following result. 1 Lemma 1: λi± has negative real part, which implies the origin of the open-loop system (5) is exponentially stable. rcr¯ 2 r¯ r 0.5 Proof: Let ∆ = ( l + µi) − 4cr¯( l + l µi). case: ∆ > 0 rcr¯ rc 0 µ < − 2¯r max l l Voltage (V) -0.5 r rcr¯ c time 0 s µmin > + 2¯r time 0.62 s l l -1 time 1.54 s time 2.53 s Using the results of Section II-A, we conclude that this case time 3.5 s rcr¯ p c -1.5 is impossible, since l < 2¯r l and the eigenvalues of the 1 2 3 4 5 Laplacian are positive (µ > 0). Nodes case: ∆ < 0 Fig. 4: N =5 RLC oscillators with consensus control rcr¯ rc µ < + 2¯r max l l results in the closed-loop system rcr¯ rc µmin > − 2¯r l l d v 0 I v = 1 1 dt a − I − L2 a The condition on µ is always satisfied since µ > 0. lc cr¯ min min | {z } Re(λ ) < 0 λ In addition, i± , and hence i± is complex and ,M1 negative. 2 Lemma 1 shows the origin of the open-loop system (5) Let ω0 = 1/(lc). According to [16, Theorem 3.1], the is stable and the oscillations decay over time. As such, con- system reaches consensus for large values of t. Specifically, sensus may not be achieved before the oscillations die out.
However, we seek to be able to sustain a rhythmic pattern 1 T 1 T vk(t) → [cos(ω0t)1 v(0) + sin(ω0t)1 a(0)] over time. In order to achieve this result, we design a voltage N ω0 feedback control that effectively cancels the resistance of each RLC, so as to emulate the behavior of LC circuits. 1 T T ak(t) → [−ω0 sin(ω0t)1 v(0) + cos(ω0t)1 a(0)], Consider the following state-feedback control, which en- N sures the individual oscillators in Fig. 3 converge to a N T where (1/N)1 = p ∈ R satisfies p ≥ 0, p L2 = 0, common solution over time, corresponding to the caterpillar T and p 1N = 1. Each oscillator evolves at its own phase and legs oscillating in synchrony. Assume a circulant topology amplitude until the network reaches a consensus (see Fig. 4). where every node is connected both to its predecessor and The consensus amplitude is a weighted average of the initial successor nodes, and the head to the tail. A feedback input conditions, and the phase is arbitrary. Matlab simulations to (5) of the form were computed using r = 0.5 Ω, c = 0.1 F , l = 0.1 H, r and r¯ = 0.5 Ω. Note the consensus phase is determined by V (t) = L v + rca (8) r¯ 2 initial conditions. To regulate the phase of the output voltages under the con- 1.5 sensus control we invoke a deCentralized Pattern Generator (dCPG). The dCPG is a rhythmic pattern generator designed 1 as a virtual LC circuit labeled k=0, that takes weak feedback from the network. Node k = 0 is connected to node k = 1. 0.5 Let r10 =r ¯ denote the coupling strength from node 0 to node 1, whereas r01 = r¯ is the coupling strength from node 1 to 0 node 0. The dynamics of node 0 are v at =1 0
Voltage (V) v 1 1 -0.5 v v¨0 + (v0 − v1) = V0(t). 2 v lc lc 3 v 4 The modified network dynamics, including the dCPG node, -1 v 5 are described using the (N + 1 )-dimensional Laplacian ma- v at =0 0 trix -1.5 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 − 0 0 0 ··· 0 0 time (s) −1 3 −1 0 0 ··· 0 −1 0 −1 −2 −1 0 ··· 0 0 Fig. 5: Evolution of N = 5 RLC oscillators with dCPG and ...... consensus feedback control ...... ˜ L2 = ...... ...... ...... Next, consider a feedback controller that achieves a stable ...... traveling wave, i.e., the output propagates around the oscil- . . . .. −1 2 −1 lator network like the traveling wave observed in caterpillar 0 −1 0 · · · −1 2 locomotion. The feedback input to (5)
T T Let v˜ = [v0, ··· , vN ] , a˜ = [a0, ··· , aN ] , as before, and l T V (t) = v + rca + L a, r˜ = diag(0, r, ··· , r) . The dynamics of the N + 1 network r¯ 2 of oscillators with dCPG becomes yields v˙ = a, and a˙ = −r/(lcr¯)L v, corresponding to the 2 d v˜ ˜ v˜ 0 ˜ closed-loop system = M + 1 ˜ V (t), (9) dt a˜ a˜ lc I where d v 0 I v = −r (10) dt a L2 0 a 0˜ I˜ lcr¯ M˜ = , | {z } 1 ˜ r˜ ˜ r˜ 1 ˜ M − lc I − lcr¯L2 − l I − cr¯L2 , 3 T V˜ (t) = [V0(t), ··· ,VN (t)] is the collection of voltage Because the network is undirected and circulant, the ˜ ˜ inputs, and 0 and I are the corresponding zero and identity Laplacian L2 is the finite-difference approximation of a matrices. 2nd-order spatial derivative. Specifically, at node k, v¨k = 2 2 Consider a feedback input similar to (8), i.e., d vk/dt and L2,kv = −vk−1 + 2vk − vk+1, where L2,k is th the k row of L2. Let xk denote the position of node k and ˜ r˜ ˜ V (t) = L2v˜ +rc ˜ a˜, v(x, t) a spatially continuous representation of v(t), e.g., for r¯ large N. We have which results in the closed-loop system 2 d ˜ ˜ 2 d v v˜ 0 I v˜ −L2,kv = vk−1 − 2vk + vk+1 ≈ ∆x = 1 1 . dx2 a˜ ˜ ˜ a˜ x=xk dt − lc I − cr¯L2 | {z } ,M2 where ∆x is the (uniform) spacing between the legs of the Under the closed-loop control with dCPG, the network caterpillar. Thus, (10) approximates the partial differential reaches consensus with node k=0. If > 0, the network wave equation 2 2 of oscillators influence the dCPG, whereas if = 0, ∂ v r 2 ∂ v 2 = ∆x 2 , (vk(t), ak(t)) converge to (v0(t), a0(t)) as t → ∞ (see Fig. ∂t lcr¯ ∂x 5), and | {z } ,γ 1 v (t) = cos(ω t)v (0) + sin(ω t)a (0) 0 0 0 2 0 0 which has solution v(x, t) = 1/2 f(x + γt) + f(x − γt) ω0 for v(x, 0) = f(x) [23]. Choosing a(x, 0) = (γ/2)(df/dx) a0(t) = −ω0 sin(ω0t)v0(0) + cos(ω0t)a0(0). yields a one-directional wave, as shown in Fig. 6. 1 t=0 s 0.9 t=0 s t=5 s 0.8 t=10 s t=15 s 0.7 t=20s t=25 s 0.6
0.5
0.4
0.3 probability density function 0.2
0.1
0 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 normalized position
Fig. 6: Propagation of wave in time
IV. EXPERIMENTAL RESULTS We fabricated an astable multivibrator (see Fig. 1) using 3D-printed microfluidic transistors and microfluidic capac- itors [21]. For the resistors, we used various lengths of the connector (Tygon microbore tubing, 06420-03, Cole Parmer, Vernon Hills, IL)). The internal resistors were three Fig. 7: Experimental set-up and multivibrator output pressure times larger than the external ones, which gave the ca- from experiment with constant input pressure 230 mbar. pacitors the opportunity to fully charge before switching. We used the MAESFLO system (Fluigent, Paris, France), which includes a Microfluidic Flow Control System and consensus and traveling wave solution in the RLC network. a FLOWELL microfluidic flow sensor, and the Sensirion A dCPG was successfully implemented as an input to the SLI-1000 flow sensor to regulate the input pressures while RLC network to regulate the collective phase. Preliminary simultaneously monitoring the flow rates through the 3D- experimental results for the microfluidic testbed are included. printed devices. We used Tygon microbore tubing and micro In ongoing work, we seek to implement the RC circuit design pipe fittings to connect the circuit components. We conducted in microfluidics and design fluidic leg actuators for a soft- all experiments at room temperature environment (20–25◦ robotic caterpillar. C). VI.ACKNOWLEDGEMENTS To examine the performance of the circuit, we connected the input point to the Fluigent system and the output to We acknowledge support of Joshua Hubbard and Ryan the flow unit. Using the Fluigent MAESFLO software, we Sochol for the microfluidic experiments. incrementally increased the pressure within the circuit. Once REFERENCES all the air bubbles were out of the circuit and the pressure stabilized at a constant value (usually around 200–250 mbar), [1] M. Wehner, R. L. Truby, D. J. Fitzgerald, B. Mosadegh, G. M. Whitesides, J. A. Lewis, and R. J. Wood, “An integrated design we collected data from the flow unit, plotted in Fig. 7. These and fabrication strategy for entirely soft, autonomous robots,” Nature, preliminary results indicate the output pressure oscillates as vol. 536, no. 7617, pp. 451–455, 2016. expected. Moreover, the results correlates that of an elec- [2] F. Saunders, E. Golden, R. D. White, and J. Rife, “Experimental verification of soft-robot gaits evolved using a lumped dynamic tronic astable multivibrator with capacitance C1 = C2 = 0.1 model,” Robotica, vol. 29, no. 6, pp. 823–830, 2011. mF, resistance R1 = R2 = 10 kΩ, and a period τ0 ≈ 0.6932 [3] R. F. Shepherd, F. Ilievski, W. Choi, S. A. Morin, A. A. Stokes, A. D. seconds. Mazzeo, X. Chen, M. Wang, and G. M. Whitesides, “Multigait soft robot,” Proceedings of the National Academy of Sciences, vol. 108, V. CONCLUSION no. 51, pp. 20400–20403, 2011. [4] D. W. Schuldt, J. Rife, and B. Trimmer, “Template for robust soft-body Caterpillar locomotion enables robust, albeit slow, naviga- crawling with reflex-triggered gripping,” Bioinspiration & biomimet- tion of irregular terrains. It is initiated from the tail to the ics, vol. 10, no. 1, p. 016018, 2015. [5] B. A. Trimmer, A. E. Takesian, B. M. Sweet, C. B. Rogers, D. C. front, and each left-right pair of legs is lifted simultaneously. Hake, and D. J. Rogers, “Caterpillar locomotion: A new model for We used RC and RLC oscillators to mimic the locomotory soft-bodied climbing and burrowing robots,” in 7th International pattern of the caterpillars as a precursor to a microfluidic Symposium on Technology and the Mine Problem, vol. 1, pp. 1–10, Monterey, CA: Mine Warfare Association, 2006. fabrication design. We characterized the locomotion gait in [6] L. van Griethuijsen and B. Trimmer, “Locomotion in caterpillars,” the RC network, and designed a closed-loop control for Biological Reviews, vol. 89, no. 3, pp. 656–670, 2014. [7] S. Vedel, Millisecond dynamics in microfluidics: Equivalent circuit [15] R. C. Dorf and J. A. Svoboda, Introduction to electric circuits. John theory and experiment. PhD thesis, DTU Nanotech, Department of Wiley & Sons, 2010. Micro and Nanotechnology, 2009. [16] W. Ren, “Synchronization of coupled harmonic oscillators with local [8] R. D. Sochol, E. Sweet, C. C. Glick, S. Venkatesh, A. Avetisyan, K. F. interaction,” Automatica, vol. 44, no. 12, pp. 3195–3200, 2008. Ekman, A. Raulinaitis, A. Tsai, A. Wienkers, K. Korner, K. Hanson, [17] W. Ren and R. W. Beard, Distributed consensus in multi-vehicle A. Long, B. J. Hightower, G. Slatton, D. C. Burnett, T. L. Massey, cooperative control. Springer, 2008. K. Iwai, L. P. Lee, K. S. J. Pister, and L. Lin, “3D printed microfluidic circuitry via multijet-based additive manufacturing,” Lab on a Chip, [18] R. Sepulchre, D. A. Paley, and N. E. Leonard, “Stabilization of planar vol. 16, pp. 668–678, 2016. collective motion: All-to-all communication,” IEEE Transactions on [9] A. S. Sedra and K. C. Smith, Microelectronic circuits theory and Automatic Control, vol. 52, no. 5, pp. 811–824, 2007. applications. Oxford University Press. [19] W. Ren, R. W. Beard, and T. W. McLain, “Coordination variables [10] F. Saunders, B. A. Trimmer, and J. Rife, “Modeling locomotion of and consensus building in multiple vehicle systems,” in Cooperative a soft-bodied arthropod using inverse dynamics,” Bioinspiration & Control, pp. 171–188, Springer, 2005. biomimetics, vol. 6, no. 1, p. 016001, 2010. [20] R. W. Beard and V. Stepanyan, “Information consensus in distributed [11] T. Umedachi and B. A. Trimmer, “Autonomous decentralized control multiple vehicle coordinated control,” in 42nd IEEE Conference on for soft-bodied caterpillar-like modular robot exploiting large and Decision and Control, vol. 2, pp. 2029–2034, IEEE, 2003. continuum deformation,” in Intelligent Robots and Systems (IROS), [21] R. D. Sochol, E. Sweet, C. C. Glick, S. Venkatesh, A. Avetisyan, K. F. 2016 IEEE/RSJ International Conference on, pp. 292–297, IEEE, Ekman, A. Raulinaitis, A. Tsai, A. Wienkers, K. Korner, K. Hanson, 2016. A. Long, B. J. Hightower, G. Slatton, D. C. Burnett, T. L. Massey, [12] T. Umedachi, K. Takeda, T. Nakagaki, R. Kobayashi, and A. Ishiguro, K. Iwai, L. P. Lee, K. S. J. Pister, and L. Lin, “3D printed microfluidic “Fully decentralized control of a soft-bodied robot inspired by true circuitry via multijet-based additive manufacturing,” Lab on a Chip, slime mold,” Biological cybernetics, vol. 102, no. 3, pp. 261–269, vol. 16, no. 4, pp. 668–678, 2016. 2010. [13] T. Vicsek and A. Zafeiris, “Collective motion,” Physics Reports, [22] K. W. Oh, K. Lee, B. Ahn, and E. P. Furlani, “Design of pressure- vol. 517, no. 3, pp. 71–140, 2012. driven microfluidic networks using electric circuit analogy,” Lab on a [14] D. A. Paley, N. E. Leonard, R. Sepulchre, D. Grunbaum, and J. K. Chip, vol. 12, no. 3, pp. 515–545, 2012. Parrish, “Oscillator models and collective motion,” IEEE Control [23] S. J. Farlow, Partial differential equations for scientists and engineers. Systems, vol. 27, no. 4, pp. 89–105, 2007. Courier Corporation, 1993.