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How wing kinematics affect power requirements and aerodynamic force production in a robotic bat wing

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Bioinspiration & Biomimetics Bioinspir. Biomim. 9 (2014) 025008 (10pp)

doi:10.1088/1748-3182/9/2/025008

How wing kinematics affect power requirements and aerodynamic force production in a robotic bat wing Joseph W Bahlman 1,3 , Sharon M Swartz 1,2 and Kenneth S Breuer 1,2 1 2

Department of Ecology and Evolutionary Biology, Brown University, Providence, RI, USA School of Engineering, Brown University, Providence, RI, USA

E-mail: [email protected], [email protected] and [email protected] Received 1 October 2013, revised 19 March 2014 Accepted for publication 21 March 2014 Published 22 May 2014 Abstract

Bats display a wide variety of behaviors that require different amounts of aerodynamic force. To control and modulate aerodynamic force, bats change wing kinematics, which, in turn, may change the power required for wing motion. There are many kinematic mechanisms that bats, and other flapping animals, can use to increase aerodynamic force, e.g. increasing wingbeat frequency or amplitude. However, we do not know if there is a difference in energetic cost between these different kinematic mechanisms. To assess the relationship between mechanical power input and aerodynamic force output across different isolated kinematic parameters, we programmed a robotic bat wing to flap over a range of kinematic parameters and measured aerodynamic force and mechanical power. We systematically varied five kinematic parameters: wingbeat frequency, wingbeat amplitude, stroke plane angle, downstroke ratio, and wing folding. Kinematic values were based on observed values from free flying Cynopterus brachyotis, the species on which the robot was based. We describe how lift, thrust, and power change with increases in each kinematic variable. We compare the power costs associated with generating additional force through the four kinematic mechanisms controlled at the shoulder, and show that all four mechanisms require approximately the same power to generate a given force. This result suggests that no single parameter offers an energetic advantage over the others. Finally, we show that retracting the wing during upstroke reduces power requirements for flapping and increases net lift production, but decreases net thrust production. These results compare well with studies performed on C. brachyotis, offering insight into natural flight kinematics. Keywords: bat, robot, flight kinematics, mechanical power, aerodynamic force (Some figures may appear in colour only in the online journal)

weight, (e.g. Jones 1972, Beasley et al 1984, Kurta and Kunz 1987). Each of these behaviors requires increasing and precisely controlling lift and/or thrust relative to steady, forward flight without load, and changes in aerodynamic force production are controlled by varying wing kinematics. A number of studies have considered how bats overcome a range of mechanical challenges (e.g. changing speeds, carrying loads, ascending, etc), and have quantified how kinematics change with flight behavior (Iriarte-D´ıaz 2009, 2012, Riskin et al 2010, MacAyeal et al 2011, Hubel et al 2012, Wolf et al

1. Introduction Bats engage in many different behaviors that require variable aerodynamic force. Depending on the bat’s ecology, a bat may perform rapid accelerations to ambush and catch insects, perform complex maneuvers through dense foliage, carry heavy loads such as food and offspring, or seasonally change 3

Present address: Department of Zoology, University of British Colombia, Vancouver, BC, Canada.

1748-3182/14/025008+10$33.00

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© 2014 IOP Publishing Ltd

Printed in the UK

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Bioinspir. Biomim. 9 (2014) 025008

2010). However, because bats vary many kinematic parameters simultaneously, these studies have not identified the individual contributions of each to aerodynamic force production. As wing kinematics change, it is likely that enegetic input to wing motion also varies. For example, thrust can be increased by increasing wingbeat frequency, but moving the wings faster also requires more mechanical power. Although bats and other flying animals can vary multiple kinematic parameters to modulate aerodynamic force, the relative energetic cost of different kinematic strategies for increasing force production is unknown. The power requirements of different kinematic mechanisms is difficult to study, both because animals vary multiple parameters simultaneously, and because directly measuring mechanical power from all the muscles that cross the shoulder joint is extremely challenging and time and labor intensive (Tobalske and Biewener 2008). In this study, we examine the power required to generate variable amounts of aerodynamic force through different kinematic mechanisms. We examine two categories of kinematic mechanisms: kinematic parameters that affect overall wing motion, specifically wingbeat frequency, wingbeat amplitude, stroke plane angle and downstroke ratio; and one kinematic parameter that affects dynamic wing shape, specifically wing folding. We use a robotic bat wing that is capable of electronically varying each of these kinematic parameters, and is instrumented to measure both mechanical power input and aerodynamic force output. While other robotic flappers have been developed and described (e.g. Sane and Dickinson 2001, Koekkoek et al 2012), ours is distinct in having the ability to dynamically fold the wing through flexing the elbow and wrist. Using this robotic tool, we describe how force output and power input change when varying each kinematic parameter. We then compare force and power across different kinematics to compare the cost of force generation with different kinematic mechanisms.

Table 1. Target values for kinematic parameters in this study. Bold value indicates baseline estimated from the model species median.

Parameter

Target values

Wingbeat frequency (Hz) Wingbeat amplitude (deg) Stroke plane angle (deg) Downstroke ratio Folding: reduction in elbow angle (deg)

1, 2, 3, 4, 5, 6, 7, 8 35, 45, 55, 65, 75, 85 50, 60, 70, 80, 90 0.40, 0.45, 0.50, 0.55 0, 40

our model species, with the exception of wingbeat frequency and wing folding, where the upper limit is set by the capability of the robot. All kinematic descriptors, as well as force and power measurements, are described in a body-fixed reference frame with the origin at the shoulder. We employ standard anatomical nomenclature, describing the up-down axis as dorsoventral (DV) and the fore-aft axis as craniocaudal (CC) (figure 1(i)). To generate specific 3D kinematics, each motor was programmed with a smooth profile of position as a function of time. For the two shoulder axes, the profiles were cosine waves, where the wave periods were determined by the target frequency and the wave amplitudes by the target wingbeat amplitude and stroke plane angle. The upstroke and downstroke were calculated separately as half waves to allow variation in downstroke ratio. Wingbeat amplitude was symmetric about the midline. For wing folding, the profile was calculated as a sine wave, with maximum wing extension at mid-downstroke and maximum folding at mid-upstroke. Wave amplitude was determined by desired amount of folding at the elbow. Any resistance to the motor’s motion, e.g. inertial load and aerodynamic forces, leads to following errors so that the realized motion deviates slightly from prescribed motion. The kinematics reported here are realized motion, measured from the motors’ encoders, and sampled at 512 Hz and 0.1 degrees resolution. Instantaneous following error was never more than three degrees (Bahlman et al 2013). From the angular position of the joints, we calculate kinematic parameters as follows:

2. Methods 2.1. Robotic model and kinematics

• wingbeat frequency ( f ): number of wingbeats per second, the reciprocal of wingbeat period; measured as the time between peak of upstroke of successive wingbeats; • wingbeat amplitude (ϕ): total angular excursion of the wing, measured at the shoulder, calculated as the vector sum of amplitude about dorsoventral and craniocaudal shoulder axes (ϕ dv, ϕ cc) (figure 1( j))   2 + ϕ2 ; ϕ= (1) ϕDV CC

We conducted experiments with an articulated robotic bat wing based on the morphology of a medium-sized fruit-eating bat, Cynopterus brachyotis (Bahlman et al 2013). The robotic wing has three degrees of freedom: the shoulder can rotate along two orthogonal axes and the wing can fold and expand by flexing/extending the elbow and ad/abducting the digits at the wrist. Each of these degrees of freedom is controlled by an independent servo motor, and collectively allows us to vary five kinematic parameters. Although dynamically controlling other wing shape variables such as camber and angle of attack is desirable, this would require actuating more joints distal in the wing, further increasing actuation complexity and wing inertia, and was beyond the scope of this study. We selected a set of baseline values that approximate medians for these parameters in the model species, C. brachyotis (Iriarte-D´ıaz 2009) (table 1 and figures 1(a)–(h)). Each kinematic parameter was systematically varied around the baseline value to reflect observed kinematic variation of

• stroke plane angle (SP): angle between the forward horizontal axis and wing’s mean path through the lateral plane (figure 1( j)), calculated as the inverse tangent of the ratio between dorsoventral and craniocaudal amplitudes:  Adv (2) SP = tan Acc where A represents the total linear excursion of the wing tip along each axis. −1

2



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Bioinspir. Biomim. 9 (2014) 025008

(a)

(b)

(c)

(d )

(e)

(f )

(g)

(h)

(i )

(j )

(k)

Figure 1. Images of robotic wing flapping through one wingbeat cycle with kinematic values closest to median values of model species: 8 Hz wingbeat frequency, 75 degrees wingbeat amplitude, 60 degrees stroke plane angle, 0.45 downstroke ratio, and 40 degrees wing folding. (a) Upper reversal point. (b) One-quarter downstroke. (c) Mid-downstroke and maximum wing extension. (c) Three-quarter downstroke. (e) Lower reversal point. ( f ) One-quarter upstroke. (g) Mid-upstroke and maximum wing folding. (h) Three-quarter upstroke. (i) Diagram illustrating the reference axes. Lift is force measured along the dorsoventral axis and thrust is force measured along the craniocaudal axis. ( j) Wingbeat amplitude (ϕ), and stroke plane angle (SP). (k) Wing folding; amount of folding is quantified as the difference in elbow angle between the maximally expanded (upper) and minimally expanded (lower) wing posture.

AXX = 2b sin



XX

2



The net force along each axis was calculated as the mean of the entire trial with no filtering of the raw signal, and values were normalized by the non-flapping (zero frequency) forces measured at the corresponding wind speed. For a few representative trials, we also calculated phase-averaged instantaneous forces. To focus on the forces associated with the primary flapping frequency, we filtered the instantaneous force data with a band pass filter at the wingbeat frequency ± 0.5 Hz (MATLAB function: butter). The structural resonant frequency of the thrust/drag measurement axis was 6.8 Hz, within the range of wingbeat frequencies we sampled. This resonance distorted the measurement signal at that frequency, and for this reason, we do not present phase-averaged data from the thrust axis at frequencies close to the resonant condition. A careful analysis of the data confirmed that, with the exception of a single data point at 7 Hz, which indicated slightly atypical behavior, the thrust/drag measurements were reliable over the entire frequency range. Power was measured as the product of motor torque and angular velocity, and summed for the two shoulder motors. Additional methodological detail can be found in Bahlman et al (2013).

(3)

where b is the wingspan; • downstroke ratio (DSR): fraction of the wingbeat duration comprising downstroke or downstroke period divided by wingbeat period, where downstroke period is calculated as time between maximum wing position at the top of upstroke and minimum wing position at the bottom of downstroke; • wing folding: reduction in span and area of wing. Because folding joints flex synchronously, we quantified folding as the difference in elbow angle between the maximally extended posture, approximately 110 degrees, and the minimally extended posture (figure 1(k)). 2.2. Performance

We conducted experiments in a low-speed, closed-circuit wind tunnel at 5 m s−1. Force and power were measured perpendicular and parallel to the wind. Flapping trials were conducted for 20 s. Complete methodology for acquiring and processing force and power data can be found in Bahlman et al (2013). 3

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3.2. Phase-averaged instantaneous power

2

2.0 Hz 4.0 Hz 5.9 Hz 7.8 Hz

1. 5

The mechanical power required to flap comprises two components: inertial and aerodynamic. A purely inertial profile is positive when the wing is accelerating and negative when the wing is decelerating. Thus, the purely inertial profile cycles at twice wingbeat frequency, with positive peaks during the first half of both upstroke and downstroke, and negative peaks during the second half of both upstroke and downstroke (figure 3(a)). The purely aerodynamic profile is positive when the wing moves opposite to the net aerodynamic force and negative when the wing moves in the same direction as the net aerodynamic force. Thus, the purely aerodynamic profile should have a positive peak at mid-upstroke and downstroke, where wing velocity and aerodynamic force are greatest (figure 3(c)). Along the DV axis, the power profiles can again be described as the sum of aerodynamic and inertial components, where the relative contribution of each varies with wingbeat frequency. At low frequencies, e.g. 2 Hz, when inertial effects were small, power peaked at mid-stroke (figure 3(e)). Power was positive during the downstroke when the wing was moving against lift and slightly negative during the upstroke when lift helped to raise the wing. At higher frequencies, e.g. 5.9 and 7.8 Hz, the wing generated negative lift during the upstroke, and the wing moved opposite to aerodynamic force during both up- and downstroke, resulting in large peaks in power during both strokes (figure 3(e)). At low frequencies, the inertial component of power was small compared to the aerodynamic component because slow wing motion generates minimal inertial force, while the cambered wing generated comparatively large aerodynamic force in the free stream flow. At higher frequencies, the inertial effects became more apparent, shifting the peaks of power closer to the beginning of each stroke where inertial effects are greatest, and power became negative at the end of each stroke during the latter part of the wing’s deceleration. During the first half of the wing’s deceleration, the relatively high aerodynamic force helped decelerate the wing, but as aerodynamic force decreased toward the end of each stroke, it was no longer sufficient to slow the wing, and the motor provided the retarding force, indicated by negative power. Mechanical power along the CC axis was a small fraction of total power in the DV axis (figures 3(e), ( f )). At 7.8 Hz, peak values along the CC axis were only 14% and average positive power only 10% of the values for the DV axis. We attribute the relatively small CC power to smaller inertial and aerodynamic costs of that axis. The inertial component was smaller because, at a stroke plane angle of 60 degrees, the wing’s CC amplitude, velocity, acceleration, and inertial cost are much less than in the DV axis. The strong positive and negative peaks during both strokes suggest inertial effects dominated the power requirements in this axis, particularly at 7.8 Hz (figure 3( f )). The aerodynamic component of power was small because along the CC axis the wing experiences both thrust and drag, which largely cancel, resulting in small net aerodynamic force and a small aerodynamic power component.

normalized lift

1 0.5 0

−0.5 −1

−1. 5 −2 0

0.2

0.4 0.6 0.8 fraction of wingbeat

1

Figure 2. Filtered phase-averaged lift forces for a range of wingbeat frequencies. Curves are mean and 95% confidence intervals for all wingbeats in a 20 s trial (N = 40 to 156). Gray area indicates downstroke.

3. Results and discussion 3.1. Phase-averaged instantaneous forces

Filtered, phase-averaged instantaneous force measurements along the dorsoventral (DV) axis showed a positive peak during downstroke, indicating lift generation, and a negative peak during the upstroke, indicating negative lift generation, for flapping frequencies of 4 Hz or greater (figure 2). Because the static wing was asymmetric, e.g. possessing positive camber and angle of incidence, it produced lift even when not flapping. Thus the magnitude of positive lift was greater than negative lift, resulting in a positive net force of 0.158 N at 7.8 Hz. The magnitudes of both positive and negative forces increased with wingbeat frequency. The force profiles included both inertial and aerodynamic contributions. At lower frequencies, where inertial effects were small, force peaked during mid-stroke, where aerodynamic force was expected to be greatest. As wingbeat frequency increased, force peaks, both positive and negative, shifted toward the wing reversal points, where inertial force was highest. The generation of negative lift during the upstroke was consistent with wake studies of live bats. The wakes of C. brachyotis, the species on which the flapper is based, showed reversed vortex pairs during upstroke, which indicate negative lift (Hubel et al 2009, 2010). Studies of Glossophaga soricina and Leptonycteris yerbabuenae also showed reversed vortex pairs during upstroke, with accompanying small negative forces (Muijres et al 2011). The negative lift we observe is, however, substantially larger, presumably because these trials were conducted with symmetric wing strokes. 4

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(a)

(b)

(c)

(d )

(e)

(f )

Figure 3. Mechanical power profiles. (a), (b) Expected contribution of aerodynamic power based on theoretical considerations.

(c), (d) Expected contribution of inertial power based on theoretical considerations. (e), ( f ) Measured total power at multiple wingbeat frequencies, including both aerodynamic and inertial components. Profiles indicate the mean of all the wingbeats in a 20 s trial. Gray region indicates the downstroke.

than upstroke. This difference is magnified as frequency increases, resulting in increasing net lift. Power increases with a leading third order dependence on frequency, which fits basic theoretical predictions (Azuma 2006). Thrust balanced drag, i.e. the fitted line crosses zero thrust, at approximately 6 Hz, slightly lower than but still close to 8.4 Hz, the median flapping frequency reported for C. brachyotis. As mentioned in section 2.2, the resonant frequency of the thrust axis was approximately 7 Hz, and it is possible that normalized thrust at 7 Hz (figure 4(a)) is slightly elevated due to this resonant interaction. The data point is retained for completeness, but should be viewed with caution. Lift, thrust, and power all increased with wingbeat amplitude, although at different rates (figures 4(b), ( f )). Approximately 60 degrees wingbeat amplitude generated enough thrust to balance drag at a wind speed of five meters per second. When varying the stroke plane angle, we see that the net lift increased slightly while net thrust increased substantially (figure 4(c)). We expected lift to decrease and thrust to increase as stroke plane angle increases toward 90 degrees, because changing the stroke plane angle changes the direction of flow

3.3. Varying kinematics that affect whole wing motion

At the baseline kinematics, (wingbeat frequency of 7.8 Hz, wingbeat amplitude of 78 degrees, stroke plane angle of 60 degrees, downstroke ratio of 0.48, 0 folding), the robot produced 0.158 N of net lift (DV force), and 0.041 N net thrust (CC force). Based on two wings, the lift would support a mass of 32 g, which falls within the range of body mass for C. brachyotis, 30–40 g (Norberg and Rayner 1987). Varying each of the four shoulder kinematic parameters produced substantial changes in net thrust and mechanical power, and smaller changes in net lift (figure 4). In all cases, power in the CC axis was a small fraction of power in the DV axis. Over the range of flapping frequencies, lift, thrust, and power all increased with frequency, although at different rates (figures 4(a), (e)). With a perfectly symmetric stroke and vertical stroke plane angle, we would expect net lift to remain constant at all frequencies because lift generated during downstroke and negative lift generated during upstroke would increase equally with wingbeat frequency, resulting in no net change in lift. However, because we used a stroke plane angle of 60 degrees, flow over the wing was faster during downstroke 5

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(a)

(b)

(c)

(d )

(e)

(f )

(g)

(h)

Figure 4. Influence of kinematics on aerodynamic force and power. All trials are based upon 20 s of data. Error bars for power are the standard error for all wingbeats in a trial. Error bars for force are calibration error, which was greater than the SE. Force has been normalized by the non-flapping value for the relevant axis. Semi-transparent bars indicate baseline kinematics.

at the wing, orienting more aerodynamic force along the CC or thrust axis than along the DV or lift axis. While the trend of thrust fits this model, that of lift does not. This may be because the flapper wing has a fixed orientation and hence a fixed angle of incidence. Angle of incidence combines with the angle of the airflow over the wing to produce an effective angle of attack. Because our wing does not twist or pronate at the shoulder as a bat’s wing would, the effective angle of attack increases with stroke plane angle. Therefore, as stroke plane angle increases, aerodynamic force orientation rotates forward, but the absolute magnitude of the force also increases, and lift is slightly higher at higher stroke plane angles. Bats appear to vary angle of incidence, and decouple angle of attack from stroke plane angle (Iriarte-D´ıaz 2009). At 90 degrees, the highest stroke plane angle tested, net thrust decreased. This decrease could simply be due to scatter in the data. Alternatively, since the effective angle of attack is highest at a stroke plane angle of 90 degrees, this may have been the point when the wing began to stall, resulting in excess drag. Stall would not be detected as a loss in net lift if both upstroke and downstroke underwent similar degrees of stall because most of the lift produced during downstroke is canceled by negative lift produced during upstroke. Power increased with stroke plane angle in the same manner as thrust (figure 4(g)). Although theoretical models predict aerodynamic power will scale with wing tip velocity (Norberg 1990, Azuma 2006), we did not observe this pattern

for stroke plane angle; average tip velocity is approximately the same for all stroke plane angles, but total mechanical power changes substantially. We attribute the change in power with increased use of the DV axis, which has a relatively high cost of motion, and decreased use of the CC axis, which has a relatively low cost of motion. Increasing the stroke plane angle is achieved by increasing the DV amplitude and decreasing the CC amplitude. In our flapper, power in the CC axis is very small compared to that in the DV axis, hence changes in total power are primarily due to changes in the DV power, which is associated with changes in DV amplitude. If we compared stroke plane and wingbeat amplitude trials that both have the same DV amplitude, they also have the same power (figures 4( f ), (g)). The decrease in power at a stroke plane angle of 90 degrees could also be explained by stall, where loss of lift would correspond to a reduction in aerodynamic power along that axis. For the downstroke ratio, both thrust and power increased as the downstroke ratio deviated from symmetry (downstroke ratio = 0.5) (figures 4(d), (h)). As downstroke ratio deviates from symmetry, one stroke velocity increases and the other decreases, while the average remains unchanged. Because aerodynamic force and power are proportional to the square and cube, respectively, of the stroke velocity, both force and power will increase more during the faster stroke than they will decrease during the slower stroke. Thus, the net result is higher force and power. 6

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(a) (c)

(b)

Figure 5. Effect of wing folding on (a) force and (b) mechanical power; solid lines/open circles indicate trials where wing remained

extended, dashed lines/filled circles indicate trials with folding during upstroke. (c) Changes in time-resolved power profiles. Gray region indicates downstroke.

also reduced wing inertia and the inertial component of power (Riskin et al 2012). Folding the wing during the upstroke and expanding the wing during the downstroke produced a number of changes in the time-resolved power profile (figure 5(c)). During upstroke, the peak power was greatly reduced: both the aerodynamic and inertial components declined, indicated by the reduction in the positive and negative peaks respectively. During downstroke, peak power was shifted later in the mid-downstroke, because mid-stroke is the point at which the wing reaches maximum expansion. Wing folding does not, however, reduce the maximum magnitude of downstroke power, although folding decreases wing area during most of downstroke. The lack of decrease in power could be explained by the increased motion of the wing tip associated with wing expansion. When the wing expanded, the leading edge distal to the wrist rotated forward, which increased tip velocity and, consequently, aerodynamic force. Force measurements with greater temporal resolution could test this hypothesis. The additional lift afforded by folding during the upstroke highlights the lift-generating potential of reducing upstroke force. With baseline kinematics, downstroke produces 0.62 N, which would support 125 g, or approximately three times the weight of the model bat species. Using more symmetric kinematics, where the upstroke mirrors the downstroke, upstroke cancels most of downstroke force. However, kinematics that reduce upstroke negative lift, such as varying angle of attack through the wingbeat cycle or employing more extreme wing folding, might increase the amount of weight a bat could carry or the speed at which a bat could fly vertically (e.g. MacAyeal et al 2011). Alternatively, reduced upstroke force could allow animals to use less energetically

3.4. Wing folding during upstroke

Folding the wing during upstroke increased net lift while decreasing net thrust and power across the range of wingbeat frequencies (figures 5(a), (b)). At 7.8 Hz, folding the wing 40 degrees at the elbow increased net lift by 50%, decreased net thrust by 40%, and decreased average power by 10% (Bahlman et al 2013). In this kinematic configuration, a two-winged model would generate enough lift to support 51 g, which is significantly more than mean body mass of C. brachyotis. The model generated enough thrust to balance drag, i.e. the fitted line crosses zero thrust, at approximately 7.5 Hz, close to this bat species’ median wingbeat frequency. Changes in force and power during wing folding were due to decreased wing area and span during the upstroke. In particular, the energetic savings from folding the wing during upstroke does not result from reducing drag, but from reducing negative lift. During upstroke, the wing generated negative lift, which canceled with positive lift generated during downstroke, providing a net lift. Reducing wing area during upstroke reduced the amount of negative lift produced, resulting in greater positive net lift. In general, the wing produced positive thrust during both up- and downstroke, which summed to total net thrust. Reducing the area during upstroke reduced overall thrust production. Power decreased due to a reduction in both the aerodynamic and inertial components of power. The aerodynamic component of power decreased because the magnitude of aerodynamic force that resisted wing motion decreased, and the shorter span reduced the moment arm of the aerodynamic force, requiring less torque to move the wing in opposition to the aerodynamic force. Reducing the span 7

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(a)

(b)

(c)

(d )

Figure 6. Power in relation to normalized net force over a range of different kinematic parameters. Graphs illustrate the power required to generate a given force; each point represents one trial from figures 4 and 5). Force is normalized by the non-flapping condition. Error bars indicate calibration error in force; error bars for power are smaller than the circles. Large, open black circle indicates baseline kinematics, and small open black circle the non-flapping state. (a), (b) Comparison among the four kinematic parameters that affect whole wing motion, i.e. actuated at shoulder, and (c), (d) effect of wing folding. Red points are the same in top and bottom.

costly motions, such as lower wingbeat frequencies, to produce required lift. The tradeoff for producing less negative lift during upstroke is producing less thrust during upstroke. Several studies have quantified substantial wing folding in C. brachyotis, as well as variable angle of attack and other kinematic flexibility that could reduce upstroke negative lift (Iriarte-D´ıaz 2009, Hubel et al 2012, Riskin et al 2012). C. brachyotis has not been observed to generate net thrust during the upstroke, as is consistent with a kinematic strategy that involves tradeoffs between negative lift and thrust (Iriarte-D´ıaz et al 2011).

curve for thrust (figure 6(d)). These results suggest that some kinematic parameters may be energetically equivalent, i.e. some sets of kinematics may require similar power cost to generate a given increase in force, and some may produce a given increase in force using a very different power cost. The four shoulder-based parameters appear to be energetically equivalent across the sampled range of kinematics, exhibiting similar power requirements to achieve similar values of lift and thrust. We note that evaluating true energetic sensitivity would require estimating differential change in power associated with a specific kinematic variable while maintaining constant lift and thrust. We were not able to accurately evaluate differential changes due to limitations in the resolution of kinematic parameters in this dataset. Nevertheless, the observation that variations in the four shoulder-based parameters generate approximately a single curve for power versus net force suggests that no one parameter is distinctly more efficient in generating incremental aerodynamic force than any other within the biologically relevant range that we sampled. This is consistent with observations of flight performance of bats, including the species on which the flapper is based, C. brachyotis. In a study in which bats carried loads of as much as 20% of body weight, individual bats varied different kinematic parameters to generate the necessary additional lift,

3.5. Comparing cost of force production

To examine cost of generating specific amounts of force through different kinematic mechanisms, we compare incremental power versus force for different kinematic variations (figure 6). When comparing the four parameters that are actuated at the shoulder, (wingbeat frequency, amplitude, stroke plane and downstroke ratio) the values for all of the trials fall on approximately a single curve in both the lift and thrust axes (figures 6(a) and (b), respectively). In contrast, comparing trials with and without wing folding, the data produce two very distinct curves for lift (figure 6(c)) and close to a single 8

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suggesting that no single kinematic strategy has a dramatic advantage over others (Iriarte-Diaz et al 2012). Additional studies required individual bats to fly at different speeds in a wind tunnel, a mechanical challenge that requires additional thrust to overcome increased body drag at higher speeds (Iriarte-D´ıaz 2009, Riskin et al 2010). These studies also showed a variety of kinematic strategies and high individual variation, a pattern consistent with absence of major energetic advantages to specific kinematic strategies. In contrast to the similarity in the power versus lift curves we observed in shoulder-based parameters (figure 6(a)), significant changes in the power versus lift curve are achieved using wing folding (figure 6(c)). The amount of lift that is generated by a constantly extended wing at 8 Hz is generated by a dynamically folding wing at around 2 Hz and requires a small fraction of the power. Thrust decreases simultaneously, so the scenarios are not completely comparable, but this suggests that bats could reduce the energetic cost of flight by combining wing folding, which increases lift, with lower wingbeat frequencies and amplitudes, which reduce lift but decrease power. In contrast, the cost of generating thrust appears to increase a small amount with wing folding. This is likely due to reduced thrust during upstroke when the wing is folded. These results show that it is possible to generate similar amounts of lift with different energetic requirements. This finding can help explain some variation seen in studies of flapping flight energetics (Engel et al 2010). Fixedwing aerodynamic theory predicts a U-shaped relationship between flight power and speed, but empirical studies have demonstrated a range of animal flight power curve shapes, including relatively flat, J-shaped, and U-shaped (e.g. Dial et al 1997, Pennycuick et al 2000, Tobalske et al 2003, Askew and Ellerby 2007). One study demonstrated curve shape variation among individuals of a single species (von Busse et al 2013). This variation implies that similar amounts of net force can be produced with using different amounts of power. Different flight styles, e.g. high upstroke force versus low upstroke force, can produce different shaped curves.

somewhat in thrust production. The differences between the two types suggest different kinematic mechanisms may vary in suitability for specific flight tasks. When maintaining or increasing thrust is important, flight can be performed at higher energetic cost by generating large amounts of negative and positive lift that cancel. This may be particularly relevant for flight at high speed and accelerating flight. Conversely, when thrust is less of a concern, flight can be achieved at lower energetic cost using wing folding. This would be the case for flight near the minimum power speed or climbing flight. Although these results demonstrate the utility of our robotic model for studying animal flight mechanics and energetics, our current flapper differs from flapping animals and, to some extent, other flapping robots in several ways. Our flapper does not have flexible digits, and therefore resists long axis twist, which is observed in bird feathers and bat fingers. Our flapper is also not capable of long axis rotation, i.e. supination and pronation, which is a characteristic of many small hovering animals such as flies, bees, and hummingbirds. Previous flapping robots modeling insect and bird flight have examined force enhancement of stroke reversal during hovering (Dickinson et al 1999, Sane and Dickinson 2001, Birch et al 2004, Altshuler et al 2005). Bat wings in forward flight exhibit little long axis rotation, which has been interpreted as dynamically varying angle of attack; this effect has been studied with a bat-like flapper that can vary angle of attack (Koekkoek et al 2012). We were not able examine long axis rotation with our flapper, and so may measure forces still smaller than what live animals can produce. Previous flappers have been actuated only at the shoulder, and the present model is a distinct advance because of its ability to dynamically control wing shape, specifically folding, during upstroke. A dynamically folding wing illustrates variation in performance, quantified as force versus power, which differs substantially from the performance variation of kinematics actuated at the shoulder (i.e. movements that affect whole wing motion). The performance effects of dynamic wing shape change is a new arena of study, and we illustrate here that a greater range of flight performance is possible when wing shape can be modulated during the wingbeat cycle. Future studies that further increase the realism and sophistication of bio-inspired flappers will include more actuated joints within the wing to produce a broader range of dynamic wing shapes, and will help map a larger performance space that further encompasses biological variation.

4. Conclusions Flapping flyers with wings that have many controllable degrees of freedom possess the potential to modulate aerodynamic forces through many different kinematic mechanisms. Some kinematic mechanisms require approximately the same amount of energy to generate additional net force, and so are energetically equivalent. In this study, these parameters are those that are actuated at the shoulder joint and primarily affect whole wing motion. For another mechanism, wing folding, which affects dynamic wing shape instead of motion, the cost of generating lift was substantially reduced, while the cost of generating thrust increased. The model-based results we report here suggest two types of strategies for increasing lift; one that affects force by changing wing velocity and one that affects force by changing wing area and/or force coefficients. These strategies differ considerably in their energetic consequences, but also

Acknowledgments The authors wish to thank Arjun Pande, Alex Carrere, Henry Bruce, Monika Mostowy, and Josh Nave for their assistance with data collection; and Cosima Schunk, Rhea von Busse, and Erika Giblin for the help in preparation of this manuscript. Support for this project was generously provided by AFOSR MURI F49620-01-1-0335 and FA9550-12-1-0210 monitored by Douglas Smith and NSF IOS 0723392 to SMS and KSB, an NSF Graduate Research Fellowship to JWB, and the Bushnell Graduate Research and Education Fund. 9

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Bioinspir. Biomim. 9 (2014) 025008

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How wing kinematics affect power requirements and aerodynamic force production in a robotic bat wing.

Bats display a wide variety of behaviors that require different amounts of aerodynamic force. To control and modulate aerodynamic force, bats change w...
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