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RESEARCH ARTICLE K i n e m a t i c s o f s w i m m i n g o f t h e m a n t a r a y : t h r e e - d i m e n s i o n a l analy sis of open-w a ter maneuv er abili ty Fr ank E. Fish 1, * , Allison Kolpas 2 , Andr e w Cr ossett 2 , M ichael A. Dudas 3 , Keith W . Moor ed 4 and Hilary Bart-Sm ith 5 ABST RA CT F or aqua tic animals, turning maneuv ers r epr esent a locomotor a ctivity tha t ma y not be confined to a single coordina te plane, making analy sis difficult, particularly in the field. T o measur e turning performance in a thr ee-dimensional spa ce for the manta r a y ( obula birostris ), a large open-w a ter swimmer, scaled s ter eo video r ecordings w er e collected. Mo v ements of the cephalic lobes, e y e and tail base w er e tr a ck ed to obtain thr ee-dimensional coordina tes. A ma thema tical analy sis w as performed on the coordina te da ta to calcula te the turning r a te and curva tur e (1/turning r adius) as a function of time by numerically es tima ting the deriva tiv e of manta tr ajectories thr ough thr ee-dimensional spa ce. Principal component analy sis w as used to pr oject the thr ee-dimensional tr ajectory onto the two-dimensional turn. Smoothing splines w er e applied to these turns. These ar e fle xible models tha t minimize a cos t function with a par ameter contr olling the balance betw een da ta fidelity and r egularity of the deriva tiv e. Da ta for 30 sequences of r a y s performing slo w , s teady turns sho w ed the highes t 20% of values for the turning r a te and smalles t 20% of turn r adii w er e 42.65–16.66 deg s 1 and 2.05–1.26 m, r espectiv ely . Such turning maneuv ers fall within the r ange of performance e xhibited by swimmers with rigid bodies. KEY W ORDS: Maneuv ering, Agility , T urning, Ster eo videogr aphy INTR ODUCT ION Uns tead y swimmin g is a vita l aspect of the locomotor r epertoir e of aqua tic animals ( W ebb, 1997, 2006; Fish and Domenici, 2015). Uns tead y motions ar e r ela ted to chang es in speed, orient a tion and dir ection (e.g. turning, a cceler a tion). By util izing self-induce d ins tabilitie s (i.e. muscular effort), anim als can pr oduce chang es in the s ta te of the center of mass (C OM) and effe ct maneuv ers. Maneuv ers ar e importa nt for ca tching pr e y , es cape fr om pr eda tors , negotia ting obs ta cles in comple x spa tial envir onmen ts, a v oiding envir onmen tal dis turbanc es, and ritualis tic displa y s and ma ting (Ho wland, 1974; W ebb , 1976; W eihs and W ebb, 1984; Domeni ci and Blak e, 1997; Mar esh et al., 2004; W alk er, 2004; W ar e et al., 2014; Fish and Hoffma n, 2015). T urnin g has been the focus of the majority of r esear ch on mane uv ers, parti cularly with r espect to la ter al or y a wing turns (i.e. r ota tion ar ound the v ertica l axis norm al to the axis of motion; W alk er, 2000; W ebb, 1994, 2006). T urning performanc e is assessed by measur ement s of maneuv er ability and agility . Maneuv er abil ity is the capabi lity to turn in a confined spa ce and is mea sur ed as the length-sp ecific r adius of the turn tr ajectory (R / L , wher e R is the r adius of the turn and L is total body length); agility is the r a te of turn measur ed as the ang ular v eloc ity ( ) ( Norbe rg and Ra yne r, 1987; W ebb, 1994; W alk er, 2000). Norber g and Ra yne r (1987) conside r ed an inv erse r ela tionship betw een maneuv er ability and agility , wher eby an anima l can turn tightly a t a lo w speed or mak e a wide turn a t hig her speeds. Ho w ev er, compar ed with large-bodi ed swimmers, swimm ers of small si ze can be hig hly maneuv er able and agile, beca use turning r adius is pr oportion al to body length and agility decr eases with incr easing size (W ebb, 1994; Fish and Nicas tr o, 2003; P arson et al., 2011). In addition, swimmers with gr ea ter fle xibility of the body ar e mo r e agile and ha v e hig her maneuv er ability than rigid-bod ied anima ls of equal size ( Fish, 2002; Fish and Nicas tr o, 2003; P arso n et al., 2011). The maj ority of s tud ies of turning performa nce by marine v ertebr a tes (e.g. fish , turtles , penguins, sea lions , dolphins) ha v e been confined to animals tha t w er e e xamine d in labor a tory or aqu arium settings ( W ebb and Ke y es, 1981; W ebb, 1983; Hui, 1985; Blak e et al., 1995; W ebb et al., 1996; Dome nici and Blak e, 1997; Ger s tner, 1999; Schr ank et al., 1999; W alk er, 2000; W ebb and F air child, 2001; Fi sh et al., 2003, 2012; Dome nici et al., 2004; Riv er a et al., 2006; Cheneval et al., 2007; Danos and Lauder , 2007; P arso n et al., 2011). In certai n cases, these s tud ies w er e able to dete rmine maxim al perform ance in term s of maneuv er abil ity and agil ity by manipula tion of the tes t envir onment or thr ough tr aini ng. Unr es tr aine d swimmin g by v ertebr a tes in na tur e is diffi cult to e xamine because of fa ctors such as infr equent observa tions, size of the anima l, and difficulty setting up r eliable r ecordi ng equipment. R ecently , the maneuv ering perform ance has been r ecorded for large ceta ceans by multi-sen sor tags and underw a ter cam er as tha t w er e mo unted on the whales (Hazen et al., 2009; W ar e et al., 2011, 2014; W ile y et al., 2011; Goldbo gen et al., 2013; W illiams et al., 2015). Ho w ev er, this ne w methodolog y has limita tions o wi ng to the inabilit y to a ccur a tely determine posit ion and v elocity , and to pla ce the animal in a fixed fr ame of r efer ence. As a r esult, measur es of maneuv er ability and agil ity ma y not ha v e the a ccur a cy of dir ect observa tion. F urthermor e, the associa tion of kinema tics of the mo bile contr ol surfa ces (e.g. fluk es, flippe rs) with the turning r adius and r a te ar e lim ited. The use of thr ee-dimens ional videogr aphy pr o vides the opport unity for enhan ced a ccur a cy and pr ecisi on to e xamine maneuv ering perform ance of aqua tic anima ls (Boisclair , 1992; Hugh es and Kelly , 1996). The swimm ing kinema tics of the body and fins of fish es ha v e been e xami ned with thr ee-di mensional vid eogr aphy for s teady swim ming and mane uv ering (Boi sclair, 1992; T ytell et al., 2008; Blevins and Lauder, 2012). Ho w ev er, these s tud ies w er e performe d in the labor a tory in conf ined spa ces. The only field s tud ies using thr ee-dimens ional video tr a cking w er e R eceiv ed 4 Ju ly 2017; Accepted 13 F ebruary 2018 1 Department of Biology, West Chester University, West Chester, PA 19383, USA. 2 Department of Mathematics, West Chester University, West Chester, PA 19383, USA. 3 Dudas’ Diving Duds, West Chester, PA 19380, USA. 4 Department of Mechanical Engineering and Mechanics, Lehigh University, Bethlehem, PA 18015, USA. 5 Department of Mechanical and Aerospace Engineering, University of Virginia, Charlottesville, VA 22904, USA. *Author for correspondence ([email protected]) F.E.F., 0000-0001-5973-3282 1 ' 2018 . Published by The C ompany of Biologi s ts Ltd | Journal of Experimenta l Biolog y (2018) 221, jeb166041. doi:1 0.1242/jeb.16 6041 J o u r n a l o f E x p e r i m e n t a l B i o l o g y

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