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Mutually Opposing Forces During Locomotion Can Eliminate the Tradeoff Between Maneuverability and Stability

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Mutually Opposing Forces During Locomotion Can Eliminate the Tradeoff Between Maneuverability and Stability Mutually opposing forces during locomotion can eliminate the tradeoff between maneuverability and stability Shahin Sefatia, Izaak D. Nevelnb, Eatai Rothc, Terence R. T. Mitchelld, James B. Snyderb, Malcolm A. MacIverb,e,f, Eric S. Fortuneg, and Noah J. Cowana,1 aDepartment of Mechanical Engineering, Johns Hopkins University, Baltimore, MD 21218; Departments of bBiomedical Engineering, eMechanical Engineering, and fNeurobiology, Northwestern University, Evanston, IL 60208; cDepartment of Biology, University of Washington, Seattle, WA 98195; dSchool of Osteopathic Medicine, Campbell University, Buies Creek, NC 27506; and gFederated Department of Biological Sciences, New Jersey Institute of Technology, Newark, NJ 07102 Edited by Daniel I. Goldman, Georgia Institute of Technology, Atlanta, GA, and accepted by the Editorial Board September 22, 2013 (received for review May 16, 2013) A surprising feature of animal locomotion is that organisms typ- locomotor strategies that are extremely stable and yet facilitate ically produce substantial forces in directions other than what is the control of extraordinary maneuvers (4, 10, 18, 19). necessary to move the animal through its environment, such as To investigate the relationship between antagonistic forces and perpendicular to, or counter to, the direction of travel. The effect locomotor control, we studied the glass knifefish, Eigenmannia of these forces has been difficult to observe because they are virescens, that hovers and rapidly changes direction while pro- often mutually opposing and therefore cancel out. Indeed, it is ducing opposing forces using a single elongated fin (Fig. 1A). likely that these forces do not contribute directly to movement but Glass knifefish, like other knifefish, generate thrust force pri- may serve an equally important role: to simplify and enhance marily through undulatory motions of an elongated anal fin (20– the control of locomotion. To test this hypothesis, we examined 22). The ribbon fin consists of 217 ± 27 downward-pointing rays a well-suited model system, the glass knifefish Eigenmannia vir- (table 4 in ref. 23; all statistics are quoted as mean ± SD unless escens, which produces mutually opposing forces during a hover- otherwise noted), with each ray independently controlled by a set ing behavior that is analogous to a hummingbird feeding from of muscles. These rays are oscillated in a plane transverse to the a moving flower. Our results and analyses, which include kine- body axis and can be coordinated to produce a wave that travels matic data from the fish, a mathematical model of its swimming longitudinally along the fin. In this study, we integrate biological dynamics, and experiments with a biomimetic robot, demonstrate experiments (Fig. 2), computational modeling, and experiments that the production and differential control of mutually opposing with a biomimetic robot (Fig. 1B and Fig. S1) to understand forces is a strategy that generates passive stabilization while simul- how the fish achieves both stability and maneuverability during taneously enhancing maneuverability. Mutually opposing forces rapid adjustments of its fore–aft position. Eigenmannia and other during locomotion are widespread across animal taxa, and these similar species of knifefish often partition their ribbon fin into results indicate that such forces can eliminate the tradeoff between two inward-counterpropagating waves (22) (Movie S1). The fin stability and maneuverability, thereby simplifying neural control. kinematics can be idealized as a pair of inward-traveling waves with parameters including oscillation frequency (f), wavelength bioinspired robotics | biomechanics (λ), and angular amplitude (θ) (Fig. 1C). We term the point where these two waves meet the “nodal point.” Although much is nimals routinely produce muscle commands that result in understood about the kinematics and mechanics of unidirectional A“antagonistic” (mutually opposing) forces during locomo- tion that either cancel out at each instant of time or average to Significance zero over each gait cycle (1–5). This may seem surprising because these antagonistic forces do not contribute to the cycle-averaged Animals often produce substantial forces in directions that do movement of the center of mass of the animal. Such antagonistic not directly contribute to movement. For example, running and forces are not only present during forward locomotion but in flying insects produce side-to-side forces as they travel for- hovering for animals such as hummingbirds, hawkmoths, and ward. These forces generally “cancel out,” and so their role electric fish; these animals produce large antagonistic forces and remains a mystery. Using a multidisciplinary approach, we exhibit extraordinary maneuverability during station keeping (6– show that mutually opposing forces can enhance both ma- 9). In this study, we demonstrate that active generation and dif- neuverability and stability at the same time, although at some ferential control of such antagonistic forces can eliminate the energetic cost. In addition to challenging the maneuverability– tradeoff between stability and maneuverability during locomotion. stability dichotomy within locomotion, our results challenge Stability is generally defined as the resistance to, and recovery the same tradeoff within the engineering of mobile robots. from, disturbances to an intended trajectory (10). Although This may inspire the exploration of a new set of strategies for maneuverability can be defined in several ways (11, 12), it is the design and control of mobile systems. perhaps most generally recognized as the relative amplitude of the control signal required to change movement direction (13). Author contributions: S.S., I.D.N., E.R., J.B.S., M.A.M., E.S.F., and N.J.C. designed research; That is, if a small change in the control amplitude effects a rapid S.S., I.D.N., and T.R.T.M. performed research; S.S. and N.J.C. contributed new reagents/ analytic tools; S.S., I.D.N., and N.J.C. analyzed data; and S.S., I.D.N., E.R., T.R.T.M., M.A.M., change in direction, the system would be considered highly ma- E.S.F., and N.J.C. wrote the paper. neuverable. The potential for a tradeoff between the resistance The authors declare no conflict of interest. to changes in direction and the ability to change direction ap- This article is a PNAS Direct Submission. D.I.G. is a guest editor invited by the Editorial pears self-evident (5, 10, 13, 14), and this tradeoff is indeed Board. considered a fundamental challenge for the engineering design 1To whom correspondence should be addressed. E-mail: [email protected]. – of airborne, submarine, and terrestrial vehicles (14 17). Many This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10. swimming, flying, and running animals, however, appear to use 1073/pnas.1309300110/-/DCSupplemental. 18798–18803 | PNAS | November 19, 2013 | vol. 110 | no. 47 www.pnas.org/cgi/doi/10.1073/pnas.1309300110 Downloaded by guest on September 27, 2021 A other replicates from all fish were quantitatively similar within and across individuals. The effect of nodal point position on the net thrust force generated by two inward-counterpropagating waves was inves- tigated using a biomimetic robot (Fig. 1B and Fig. S1)anda fi B computational model (Materials and Methods). In the rst set of experiments with the biomimetic robot, the nodal point position was varied while other properties of the traveling waves were held constant. Thrust forces generated by the two traveling waves were also predicted numerically. The measured forces as a func- tion of nodal point shift closely match simulated forces from our C model (Fig. 4A). The thrust force varied linearly as a function of x nodal point shift. We define the nodal point shift gain, κ,asthe y z ratio of the measured net force to the nodal point shift: Nodal point κ = FThrust : [1] ΔL r Wave motion This parameter indicates the change in force given a unit change Wave motion in nodal point position, and it is used as a metric for fore–aft maneuverability of counterpropagating waves. Note that the nodal Fig. 1. Three testbeds considered in this paper include the glass knifefish, shift gain, κ, increases as a function of frequency (f) and angular a biomimetic robot, and a model of the swimming dynamics. (A)Theglass amplitude of counterpropagating waves (θ)(Fig. S5). κ increases knifefish E. virescens. Experiments with a biomimetic robot match force approximately quadratically with the frequency (Fig. 4B). measurements predicted by a computational model of ribbon fin propulsion. fi fi We also discovered that passive damping emerges with coun- (B) A biomimetic robot that has a ventral ribbon n to emulate the nof terpropagating waves. Specifically, a damping force opposing the knifefish. The biomimetic robotic fin consists of 32 independently controlled rays, allowing for a wide range of fin kinematics, such as counterpropagating direction of velocity perturbations increases linearly as a function waves. (C) Fin is modeled as a pair of inward-traveling waves. Directions of head of the speed of the animal relative to the flow. To measure this and tail waves and kinematics of the ribbon fin are shown in this schematic: drag-like term in the robotic setup, the nodal point was held at fl θ λ angular de ection ( ), wavelength ( ),lengthsofthetwowaves(Lhead and Ltail), the center of the robotic fin ðΔL = 0Þ, making the length of the length of whole fin(Lfi ), temporal frequency (f), and nodal point (red circle). n fin dedicated to the tail wave (Ltail) identical to the length of fin dedicated to the head wave (Lhead). The measured forces pro- duced by the biomimetic robot vary linearly as a function of steady- traveling waves in a fluid (20, 24–28), far less is known (22, 29) state ambient flow and closely match simulated forces from our about counterpropagating waves, particularly in relation model (Fig. 4C). Here, we define the damping constant, β,asthe to control. ratio of the measured damping force, F,totheflow speed, U: Results F Using high-speed videography at 100 frames per second, the β = − Damping : [2] kinematics of the ribbon finoffive fish were digitized during U fi station keeping (Fig.
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