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Microfluidic Circuit Dynamics and Control for Caterpillar-Inspired Locomotion in a Soft Robot

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Microfluidic Circuit Dynamics and Control for Caterpillar-Inspired Locomotion in a Soft Robot 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 2 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) 2 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 2 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 2 3 ongoing work with the RC microfluidic network. Section V 1 −1 0 ··· 0 summarizes the paper and our future work. 6−1 2 −1 0 ··· 0 7 6 7 6 0 −1 2 −1 0 ··· 0 7 6 7 6 . .. .. .. .. 7 II. BACKGROUND 6 . 7 N×N L1 = 6 7 2 R ; 6 . .. .. .. .. 7 The proposed work models each leg of a caterpillar as 6 . 7 6 7 an RLC (or RC) circuit. The network of N caterpillar legs 6 . 7 4 . .. −1 2 −15 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 2 −1 0 · · · −13 first reviews RLC and RC circuits, followed by a background 6−1 2 −1 0 ··· 0 7 overview of graph theory, consensus dynamics, and the field 6 7 6 0 −1 2 −1 0 ··· 0 7 of microfluidics.
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