Multi-functional foot use during running in the zebra-tailed lizard (Callisaurus draconoides)
LLi et al. (2012),
The Journal of Experimental Biology , , 3293–3308. doi:10.1242/jeb.061937 1 Multi-functional foot use during running in the zebra-tailed lizard ( Callisaurus draconoides ) Chen Li , S. Tonia Hsieh , and Daniel I. Goldman School of Physics, Georgia Institute of Technology, Atlanta, GA 30332, USA and Department of Biology, Temple University, Philadelphia, PA 19122, USA * Author for correspondence ([email protected]) 6
Summary A diversity of animals that run on solid, level, flat, non-slip surfaces appear to bounce on their legs; elastic elements in the limbs can store and return energy during each step. The mechanics and energetics of running in natural terrain, particularly on surfaces that can yield and flow under stress, is less understood. The zebra-tailed lizard ( Callisaurus draconoides ), a small desert generalist with a large, elongate, tendinous hind foot, runs rapidly across a variety of natural substrates. We use high speed video to obtain detailed three-dimensional running kinematics on solid and granular surfaces to reveal how leg, foot, and substrate mechanics contribute to its high locomotor performance. Running at ~ 10 body length/s (~ 1 m/s), the center of mass oscillates like a spring-mass system on both substrates, with only 15% reduction in stride length on the granular surface. On the solid surface, a strut-spring model of the hind limb reveals that the hind foot saves about 40% of the mechanical work needed per step, significant for the lizard’s small size. On the granular surface, a penetration force model and hypothesized subsurface foot rotation indicates that the hind foot paddles through fluidized granular medium, and that the energy lost per step during irreversible deformation of the substrate does not differ from the reduction in the mechanical energy of the center of mass. The upper hind leg muscles must perform three times as much mechanical work on the granular surface as on the solid surface to compensate for the greater energy lost within the foot and to the substrate.
25 Key words: terrestrial locomotion, mechanics, energetics, kinematics, spring-mass system, elastic 26 energy savings, dissipation, granular media 27 Running title: Substrate effects on foot use in lizards 28 29 i et al. (2012),
The Journal of Experimental Biology , , 3293–3308. doi:10.1242/jeb.061937 2 Introduction
30 Rapid locomotion like running and hopping can be modeled as a spring-mass system bouncing in 31 the sagittal plane (i.e., the Spring-Loaded Inverted Pendulum model, SLIP) (Blickhan, 1989). 32 This has been demonstrated in a variety of animals (Blickhan and Full, 1993; Holmes et al., 2006) 33 in the laboratory on rigid, level, flat, non-slip surfaces (hereafter referred to as “solid surfaces”) 34 such as running tracks and treadmills (Dickinson et al., 2000). In the SLIP model, the animal 35 body (represented by the center of mass, CoM) bounces on a single leg (represented by a spring) 36 like a pogo stick, and exerts point contact on the solid ground. The leg spring compresses during 37 the first half of stance, and then recoils during the second half of stance. Through this process, the 38 mechanical (i.e., kinetic plus gravitational potential) energy of the CoM is exchanged with elastic 39 energy stored in the compressed leg spring, reducing energy use during each step. For animals 40 like insects (e.g., Schmitt et al., 2002) and reptiles (e.g., Chen et al., 2006) that run with a 41 sprawled limb posture, the CoM also oscillates substantially in the horizontal plane in a similar 42 fashion, which can also be modeled as a spring-mass system bouncing in the horizontal plane (i.e., 43 the Lateral Leg Spring model, LLS) (Schmitt et al., 2002). Both the SLIP and the LLS models 44 predict that the mechanical energy of the CoM is lowest at mid-stance and highest during aerial 45 phase.
46 In these models, the spring-mass system and the interaction with the solid ground are perfectly 47 elastic and do not dissipate energy; thus no net work is performed. However, as animals move 48 across natural surfaces, energy is dissipated both within their body and limbs (Fung, 1993) and to 49 the environment (Dickinson et al., 2000). Therefore, mechanisms to reduce energy loss during 50 locomotion can be important. The limbs of many organisms possess elastic elements such as 51 tendons and ligaments that can function as springs to store and return energy during rapid 52 locomotion like running and hopping to decrease energetic cost (Alexander, 2003). Most notable 53 for this function are the ankle extensor tendons in the lower hind leg and the digital flexor 54 tendons and ligaments in the lower fore leg (Alexander, 2003). Furthermore, different limb-55 ground interaction strategies may be utilized depending on the dissipative properties of the 56 substrate. 57 Laboratory experiments have begun to reveal mechanisms of organisms running on non-solid 58 surbstrates, such as elastic (Ferris et al., 1998; Spence et al., 2010), damped (Moritz and Farley, 59 2003), inclined (Roberts et al., 1997), or uneven (Daley and Biewener, 2006; Sponberg and Full, 60 2008) surfaces; surfaces with few footholds (Spagna et al., 2007); and the surface of water 61 (Glasheen and McMahon 1996a; Hsieh, 2003). While spring-mass-like CoM motion was 62 i et al. (2012),
The Journal of Experimental Biology , , 3293–3308. doi:10.1242/jeb.061937 3 observed only in some of these studies (Ferris et al., 1998; Moritz and Farley, 2003; Spence et al., 63 2010), a common finding is that on non-solid surfaces limbs do not necessarily behave like 64 springs to save energy. In addition, these studies suggest that both the active control of body and 65 limb movement through the nervous system, and the passive mechanical responses of 66 viscoelastic limbs and feet with the environment, play important roles in the control of rapid 67 terrestrial locomotion (for reviews, see Full and Koditschek, 1999; Dickinson et al., 2000). 68 Many substrates found in nature, such as sand, gravel, rubble, dirt, soil, mud, and debris, can 69 yield and flow under stress during locomotion and experience solid-fluid transitions, through 70 which energy may be dissipated via plastic deformation. Understanding locomotion on these 71 substrates is challenging in part because, unlike for flying and swimming where the fluid flows 72 and forces can in principle be determined by solving the Navier-Stokes equations in the presence 73 of moving boundary conditions (Vogel, 1996), no comprehensive force models yet exist for 74 terrestrial substrates that yield and flow (hereafter referred to as “flowing substrates”). 75 Granular materials (Nedderman, 1992) like desert sand which are composed of similarly sized 76 particles provide a good model substrate for studying locomotion on flowing substrates. 77 Compared to other flowing substrates, granular materials are relatively simple and the intrusion 78 forces within them can be modeled empirically (Hill et al., 2005). Their mechanical properties 79 can also be precisely and repeatedly controlled using a fluidized bed (Li et al., 2009). In addition, 80 locomotion on granular surfaces is directly relevant for many desert-dwelling reptiles and 81 arthropods such as lizards, snakes, and insects (Mosauer, 1932; Crawford, 1981). Recent 82 advances in the understanding of force and flow laws in granular materials subject to localized 83 intrusion (Hill et al., 2005; Katsuragi and Durian, 2007; Gravish et al., 2010; Ding et al., 2011a) 84 begin to provide insight into the mechanics of locomotion on (and within) granular substrates (Li 85 et al., 2009; Maladen et al., 2009; Mazouchova et al., 2010; Li et al., 2010b; Maladen et al., 2011; 86 Ding et al., 2011b; Li et al., in press). 87 The zebra-tailed lizard ( Callisaurus draconoides , SVL ~ 10 cm, mass ~ 10 g, Fig. 1A) is an 88 excellent model organism for studying running on natural surfaces, because of its high locomotor 89 performance over diverse terrain. As a desert generalist, this lizard lives in a range of desert 90 habitats including flat land, washes, and sand dunes (Vitt and Ohmart, 1977; Korff and McHenry, 91 2011), and encounters a large variety of substrates ranging from rocks, gravel, closely-packed 92 coarse sand, and loosely-packed fine sand (Karasov and Anderson, 1998; Korff and McHenry, 93 2011). The zebra-tailed lizard is the fastest-running species among desert lizards of similar size 94 i et al. (2012),
The Journal of Experimental Biology , , 3293–3308. doi:10.1242/jeb.061937 4 (Irschick and Jayne, 1999a), and has been observed to run at up to 4 m/s (50 bl/s) both on solid 95 (e.g., treadmill) (Irschick and Jayne, 1999a) and on granular (e.g., sand dunes) (Irschick and 96 Jayne, 1999b) surfaces. Its maximal acceleration and running speed also did not differ 97 significantly when substrate changes from coarse wash sand to fine dune sand, whose yield 98 strengths differ by a factor of three (Korff and McHenry, 2011). 99 Of particular interest is whether and how the zebra-tailed lizard’s large, elongate hind foot 100 contributes to its high locomotor capacity. In addition to a slim body, a long tapering tail, and 101 slender legs (Fig. 1A), the zebra-tailed lizard has an extremely large, elongate hind foot, the 102 largest (40% SVL) among lizards of similar size (Irschick and Jayne, 1999a). Its hind foot is 103 substantially larger than the fore foot (area = 1 cm vs. = 0.3 cm ) and likely plays a dominant role 104 for locomotion (Mosauer, 1932). Recent studies in insects, spiders, and geckos (Jindrich and Full, 105 1999; Antumn et al., 2000; Dudek and Full, 2006; Spagna et al., 2007) suggested that animals can 106 rely on appropriate morphology and material properties of their bodies and limbs to accommodate 107 variable, uncertain conditions during locomotion. Despite suggestions that the large foot area 108 (Mosauer, 1932) and increased stride length via elongate toes may confer locomotor advantages 109 (Irschick and Jayne, 1999a), the mechanisms of how the hind foot contributes to the zebra-tailed 110 lizard’s high running capacity remain unknown. 111 In this paper, we study the mechanics and mechanical energetics of the zebra-tailed lizard running 112 on two well-defined model surfaces, a solid surface and a granular surface. These two surfaces lie 113 on opposite ends of the spectrum of substrates that the zebra-tailed lizard encounters in its natural 114 environment, and present distinct conditions for locomotion. We investigate whether the lizard’s 115 center of mass bounces like a spring-mass system during running on both solid and granular 116 surfaces. We combine measurements of three-dimensional kinematics of the lizard’s body, hind 117 limb, and hind foot, dissection and resilience measurements of the hind limb, and modeling of 118 foot-ground interactions on both substrates, and demonstrate that the lizard’s large, elongate hind 119 foot serves different functions during running on solid and granular surfaces. We find that on the 120 solid surface, the hind foot functions as an energy-saving spring; on the granular surface, it 121 functions as a dissipative, force-generating paddle to generate sufficient lift during each step. The 122 larger energy dissipation to the substrate and within the foot during running on the granular 123 surface must be compensated for by greater mechanical work done by the upper hind leg muscles. 124 125 Materials and methods
126 i et al. (2012),
The Journal of Experimental Biology , , 3293–3308. doi:10.1242/jeb.061937 5 Animals
127 Seven adult zebra-tailed lizards (
Callisaurus draconoides ) were collected from the Mojave Desert, 128 AZ, USA in 08/2007 (Permit SP591773) for three-dimensional kinematics experiments. Table 1 129 shows the morphological measurements for these seven animals. Eleven additional adult animals 130 were collected from the Mojave Desert, CA, USA in 09/2009 (Permit SC 10901) for hind limb 131 resilience measurements. Two preserved specimens were used for dissection. The animals were 132 housed in the Physiological Research Laboratory animal facility of The Georgia Institute of 133 Technology. Each animal was housed individually in an aquarium filled with sand, and fed 134 crickets and mealworms dusted with vitamin and calcium supplement two to three times a week. 135 The ambient temperature was maintained at 28°C during the day and 24°C during the night. Full-136 spectrum fluorescent bulbs high in UVB were set to a 12 hour/12 hour light/dark schedule. 137 Ceramic heating elements were provided 24 hours a day to allow the animals to thermo-regulate 138 at preferred body temperature. All experimental procedures were conducted in accordance with 139 The Georgia Institute of Technology IACUC protocols. 140
Surface treatments
141 A wood board (120 × 23 × 1 cm ) bonded with sandpaper (grit size ~ 0.1 mm) for enhanced 142 traction was used as the solid surface. Glass particles (diameter = 0.27 ± 0.04 mm mean ± 1 143 standard deviation, density = 2.5 × 10 kg/m , Jaygo Incorporated, Union, NJ, USA) were used as 144 the granular substrate, which are approximately spherical and of similar size to typical desert sand 145 (Dickinson and Ward, 1994). Before each trial, a custom-made fluidized bed trackway (200 cm 146 long, 50 cm wide) prepared the granular substrate (12 cm deep) into a loosely packed state 147 (volume fraction = 0.58) for repeatable yield strength (for experimental details of the fluidized 148 bed trackway, see Li et al., 2009). 149 Three-dimensional kinematics
150 We used high speed video to obtain three-dimensional kinematics as the lizard ran across the 151 prepared surfaces (Fig. 1B). Before each session, high-contrast markers (Wite-Out, Garden Grove, 152 CA, USA) were painted on each animal for digitizing at nine joints along the midline of the trunk 153 and the right hind limb (Fig. 1A,B): neck (N), center of mass (CoM), pelvis (P), hip (H), knee (K), 154 ankle (A); and the metatarsal-phalangeal joint (MP), distal end of the proximal phalanx (PP), and 155 digit tip (T) of the fourth toe. The approximate longitudinal location of the CoM in resting 156 position was determined by tying a thread around the body of an anesthetized lizard and 157 i et al. (2012),
The Journal of Experimental Biology , , 3293–3308. doi:10.1242/jeb.061937 6 repositioning the thread until the body balanced horizontally. Before each trial, the surface was 158 prepared (for the granular surface treatment only), and calibration images were taken of a custom-159 made 39-point calibration object (composed of LEGO, Billund, Denmark). The animal was then 160 induced to run across the field of view by a slight tap or pinch on the tail. Two synchronized AOS 161 high speed cameras (AOS Technologies, Baden Daettwil, Switzerland) captured simultaneous 162 dorsal and lateral views at 500 frame/s (shutter time = 300 s). The ambient temperature was 163 maintained at 35°C during the test. Animals were allowed to rest at least five minutes between 164 trials and at least two days between sessions. 165 We digitized the calibration images and high speed videos, and used direct linear transformation 166 (DLT) to reconstruct three-dimensional kinematics from the two-dimensional kinematics from 167 both dorsal and lateral views. Digitization and DLT calculations were performed using custom 168 software (DLTcal5 and DLTdv5, Hedrick, 2008). Axes were set such that + x pointed in the 169 direction of forward motion, + z pointed vertically upward, and + y pointed to the left of the animal. 170 Footfall patterns of touchdown and takeoff were determined from the videos. On the granular 171 surface, because the hind foot often remained obscured by splashed grains during foot extraction, 172 we defined foot takeoff as when the knee began to flex following extension during limb 173 protraction (which is when foot takeoff occurs on the solid surface). To reduce noise and enable 174 direct comparisons among different running trials, position data were filtered with a Butterworth 175 low-pass filter with a cutoff frequency of 75 Hz, and interpolated to 0 T ) between two successive touchdowns of the right hind limb. All data analysis was 177 completed with MATLAB (MathWorks, Natick, MA, USA) unless otherwise specified. 178 Statistics
179 We accepted trials that met the following criteria: the animal ran continuously through the field of 180 view, the run was straight without contacting sidewalls of the trackway, there was a full stride 181 (between two consecutive touchdowns of the right hind limb) in the range of view, all the nine 182 markers were visible throughout the full stride, and the forward speed changed less than 20% 183 after the full stride. With these criteria, out of a total of 125 trials from 7 individuals on both solid 184 (61 trials from 7 individuals) and granular (64 trials from 7 individuals) surfaces collected over a 185 period of over three months, we ultimately accepted 51 runs from 7 individuals on solid (23 runs 186 from 7 individuals) and granular (28 runs from 7 individuals) surfaces. Because the data set had 187 an unequal number of runs per individuals, and because we were measuring freely-running 188 animals and did not control for speed, to maintain statistical power, all statistical tests were 189 i et al. (2012),
The Journal of Experimental Biology , , 3293–3308. doi:10.1242/jeb.061937 7 performed on a subset of these data using one representative run per individual on both solid ( N = 190 7) and granular ( N = 7) surfaces. The representative run for each individual was selected based on 191 having the most consistent running speed for at least one full stride and was also closest to the 192 mean running speed of all 51 trials. Data are reported as mean ± 1 standard deviation (s.d.) from 193 the 7 representative runs on each substrate unless otherwise specified. 194 To determine the effect of substrate, all kinematic variables were corrected for size-related 195 differences by regressing the variables against SVL and taking the residuals for those that 196 regressed significantly with SVL ( P < 0.05). We then ran an ANCOVA with substrate and speed 197 as covariates to test for substrate effects, independent of running speeds. All statistical tests were 198 performed using JMP (SAS, Cary, NC, USA). 199 For the energetics data, we used dimensionless quantities by normalizing energies of each run to 200 the CoM mechanical energy at touchdown of that run, thus eliminating the effect of mass and 201 running speed on energies. An ANOVA was used to test the differences between the reduction in 202 CoM mechanical energy, elastic energies, and energy loss. A Tukey’s HSD was used for post-hoc 203 tests where needed. 204 Dissection and model of hind limb
205 To gain insight into the role of anatomical components of the hind limb on mechanics during 206 locomotion, we dissected the hind limb of two preserved specimens. We quantified anatomical 207 dimensions by measuring the radii of the knee (K), ankle (A), the metatarsal-phalangeal joint 208 (MP), the distal end of the proximal phalanx (PP), and the digit tip (T) of the fourth toe. We also 209 observed the muscle and tendon arrangements within the lower leg and the foot. Based on these 210 anatomical features, we developed a model of the hind limb which incorporated the structure, 211 properties, and function of its main elements. 212
Resilience measurements of hind limb
213 To characterize the resilience of the hind limb for estimation of energy return, a modification of 214 the work loop technique was used (Fig. 2A), in which the limb was kept intact and forces were 215 applied to the whole limb instead of a single muscle (Dudek and Full, 2006). The animal was 216 anesthetized using 2% isoflurane air solution during the test. The hind foot was maintained within 217 the vertical plane, pushed down onto and then extracted from a custom force platform suited for 218 small animals (10.2 × 7.6 cm , range = 2.5 N, resolution = 0.005 N) bonded with sandpaper (grit 219 size ~ 0.1 mm). Ground reaction force F was measured at 10 kHz sampling rate using a custom 220 i et al. (2012), The Journal of Experimental Biology , , 3293–3308. doi:10.1242/jeb.061937 8 LabVIEW program (National Instruments, Austin, TX, USA). A Phantom high speed camera 221 (Vision Research, Wayne, NJ, USA) simultaneously recorded deformation of the foot from the 222 side view at 250 frame/s (shutter time = 500 s). High-contrast markers (Wite-Out, Garden Grove, 223 CA, USA) were painted on the joints of the hind foot (A, MP, PP, T, and a point on the tibia 224 above the ankle). The ambient temperature was maintained at 35°C during the test. 225 Videos of foot deformation were digitized to obtain the angular displacement of the foot = 226 t , i.e., the change in the angle formed by the tibia and the foot (from the ankle to the digit tip 227 of the fourth toe) (Fig. 2A). Angular displacement was synchronized with the measured torque 228 about the ankle (calculated from the measured ground reaction force) to obtain a passive work 229 loop. The damping ratio of the hind limb, i.e., the percentage of energy lost within the hind limb 230 after loading and unloading, was calculated as the fraction of area within a work loop relative to 231 the area under the higher loading curve (Fung, 1993). Hind limb resilience, i.e., the percentage of 232 energy returned by the foot after loading and unloading, was one minus the damping ratio (Ker et 233 al., 1987; Dudek and Full, 2006). An ANOVA was used to test the effect of maximal torque, 234 maximal angular displacement, loading rate, and individual animal on hind limb resilience. 235 Granular penetration force measurements
236 While comprehensive force models are still lacking to calculate ground reaction forces during 237 locomotion granular media, a low speed penetration force model was previously used to explain 238 the locomotor performance of a legged robot on granular media (Li et al., 2009). Similarly, to 239 estimate the vertical ground reaction force on the lizard foot during running on the granular 240 surface, we measured the vertical force on a plate slowly penetrating vertically into the granular 241 substrate (Fig. 2B). Before each trial, a fluidized bed (area = 24 × 22 cm ) prepared the granular 242 substrate (depth = 12 cm) into a loosely packed state (volume fraction = 0.58) (for details, see 243 Maladen et al., 2009). A robotic arm (CRS robotics, Burlington, OT, Canada) pushed a 244 horizontally-oriented plate vertically downward at 0.01 m/s into the granular substrate to a depth 245 of 7.6 cm, and then extracted the plate vertically at 0.01 m/s. The force on the plate was measured 246 by a force transducer (ATI Industrial Automation, Apex, NC, USA) mounted between the robotic 247 arm and the plate at 100 Hz sampling rate using a custom LabVIEW program (National 248 Instruments, Austin, TX, USA). The depth of the plate was measured by tracking the position of 249 an LED light marker mounted on the robotic arm in side view videos taken by a Pike high speed 250 camera (Edmund Optics, Barrington, NJ, USA). Two thin aluminum plates of different area were 251 i et al. (2012), The Journal of Experimental Biology , , 3293–3308. doi:10.1242/jeb.061937 9 used ( A = 7.6 × 2.5 cm and A = 3.8 × 2.5 cm ; thickness = 0.6 cm). Three trials were performed 252 for each plate. 253 254 Results
Performance and gait
256 On both solid and granular surfaces, the zebra-tailed lizard ran with a diagonal gait, a sprawled 257 limb posture, and lateral trunk bending (see Fig. 3 and Movies 1, 2 in supplementary material for 258 representative runs on both substrates). Figure 4 shows average forward speed v (cid:3364) x ,CoM , stride 259 frequency f , and duty factor D of the entire data set (all symbols; 23 runs on the solid surface and 260 28 runs on the granular surface) and of the representative runs (filled symbols; N = 7 on the solid 261 surface and N = 7 on the granular surface). Table 2 lists mean values and statistical results for all 262 the gait and kinematic variables from the representative runs for both solid ( N = 7) and granular 263 ( N = 7) surfaces. On both surfaces, v (cid:3364) x ,CoM increased with f (Fig. 4A, P < 0.05, ANCOVA), and D
264 decreased with v (cid:3364) x ,CoM (Fig. 4B, P < 0.05, ANCOVA). D ≈ T between alternating stances (Fig. 5A). Neither 266 v (cid:3364) x ,CoM ( P > 0.05, ANOVA) nor D ( P > 0.05, ANCOVA) significantly differ between surfaces. 267 Average stride length = v (cid:3364) x ,CoM / f was 15% shorter on the granular surface ( P < 0.05, ANCOVA). 268 Center of mass kinematics
269 The lizard displayed qualitatively similar center of mass oscillations during running on both 270 surfaces. The CoM forward speed v x ,CoM (Fig. 5B) and vertical position z CoM (Fig. 5C) oscillated at 271 2 f , dropping during the first half and rising during the second half of a stance, i.e., reaching 272 minimum at mid-stance and maximum during the aerial phase. The CoM also oscillated medio-273 laterally at f (Fig. 5D). Throughout the entire stride, z CoM was significantly higher on the solid 274 surface ( P < 0.05, ANCOVA). The CoM vertical oscillations z CoM and lateral oscillations y CoM
275 did not differ between substrates ( P > 0.05, ANCOVA). 276 Hind foot, hind leg, and trunk kinematics
277 The lizard displayed distinctly different hind foot, hind leg, and trunk kinematics during running 278 on solid and granular surfaces (Figs. 3, 6). On the solid surface, the lizard used a digitigrade foot 279 posture (Fig. 3A E, solid line/curve). During the entire stride, the hind foot engaged the solid 280 i et al. (2012),
The Journal of Experimental Biology , , 3293–3308. doi:10.1242/jeb.061937 10 surface only with the digit tips. At touchdown, the toes were straight and pointed slightly 281 downward. The touchdown foot angle (measured along the fourth toe) was touchdown = 12 ± 4° 282 relative to the surface (Fig. 3A,E; Fig. 6A, red). During stance, the long toes pivoted over the 283 stationary digit tips (Fig. 3A C, vertical dotted line shows zero displacement) and hyperextended 284 into a c-shape (Fig. 3B, solid curve). The foot straightened again at takeoff, pointing downward 285 and slightly backward (Fig. 3C, solid line), and then flexed during swing (Fig. 3D, solid curve). 286 On the granular surface, the lizard used a plantigrade foot posture (Fig. 3F,J, solid line). At 287 touchdown, the hind foot was nearly parallel with the surface, with the toes spread out and held 288 straight. In the vertical direction, the foot impacted the granular surface at speeds of up to 1 m/s. 289 The ankle joint slowed down to ~ 0.1 m/s within a few milliseconds following impact (a few 290 percent of stride period T ) while the the foot started penetrating the surface. The touchdown foot 291 angle was touchdown = 4 ± 3° relative to the surface (Fig. 3J; Fig. 6A, blue), significantly smaller 292 than that on the solid surface ( P < 0.05, ANCOVA). During stance, the entire foot moved 293 subsurface and was obscured (Fig. 3G). The ankle joint remained visible right above the surface 294 and moved forward by about a foot length (Fig. 3F H, vertical dotted line shows ankle 295 displacement). The foot was extracted from the substrate at takeoff, pointing downward and 296 slightly backward, and then flexed during swing (Fig. 3I, solid curve). 297 As a result of foot penetration on the granular surface, both the knee height z knee (Fig. 6B) and 298 pelvis height z pelvis (Fig. 6C) were lower on the granular surface ( P < 0.05, ANCOVA). In addition, 299 on the granular surface, the knee moved downward by a larger vertical displacement z knee during 300 the first half of stance ( P < 0.05, ANCOVA; Fig. 6B), while the knee joint extended by a larger 301 angle knee during the second half of stance ( P < 0.05, ANCOVA; Fig. 6D). Throughout the 302 entire stride, the trunk was nearly horizontal on the solid surface (Fig. 3A D, dashed line), but 303 pitched head-up on the granular surface (Fig. 3F I, dashed line; Fig. 6E). On both surfaces, the 304 hind legs were sprawled at an angle of sprawl ≈
40° during stance (Fig. 3; sprawl is defined as the 305 angle between the horizontal plane and the leg orientation in the posterior view). In most runs, the 306 tail was farther from the solid surface and closer to the granular surface (Fig. 3). 307 Hind limb anatomy
308 From morphological measurements (Table 1), the hind foot of the zebra-tailed lizard comprised 309 42% of the hind limb length, and the longest fourth toe alone accounted for 63% of the hind foot 310 length. These ratios are in similar range to previous observations (Irschick and Jayne, 1999a). The 311 i et al. (2012),
The Journal of Experimental Biology , , 3293–3308. doi:10.1242/jeb.061937 11 slender foot had a cross-sectional radius of r = 0.50 r K = r A = 1.25 mm, r MP = 0.75 mm, r PP = r T = 0.50 mm. 313 Unlike many cursorial mammals whose ankle extensor muscles of the lower hind leg have long 314 tendons (Alexander, 2003), ankle extensor tendons are nearly non-existent in the zebra-tailed 315 lizard (Fig. 7A). Instead, layers of elongate tendons were found in both the dorsal and ventral 316 surfaces of the foot. Our anatomical description is focused on the ventral muscle and tendon 317 anatomy in the hind limb and terms given to muscles and tendons follow (Russell, 1993). A large, 318 tendinous sheath, the superficial femoral aponeurosis, originates from the femoro-tibial 319 gastrocnemius, stretches across the ventral surface of the foot, and inserts on the metatarsal-320 phalangeal joints for digits III and IV. The superficial portion of the femoro-tibial gastrocnemius 321 muscle body extends to the base of the ankle, thereby rendering the human equivalent of the 322 ankle extensor tendons (i.e., the “Achilles” tendon) absent. Deep to the superficial femoral 323 aponeurosis lie the flexor digitorum brevis muscles (not shown) which control the flexion of each 324 of the digits. Tendons from the flexor digitorum longus muscle located on the lower hind leg run 325 deep to the flexor digitorum brevis muscle bodies, and extend to the tips of the digits. No 326 additional tendons are visible deep to the flexor digitorum longus tendons. 327 Hind limb model
328 Based on the observed muscle and tendon anatomy, we propose a two-dimensional strut-spring 329 model of the hind limb (Fig. 7B), which assumes isometric contraction for the lower leg muscles 330 and incorporates the spring nature of the foot tendons. This model is inspired from previous 331 observations in large running and hopping animals of the strut-like function of ankle extensor 332 muscles (Biewener, 1998a; Roberts et al., 1997) and spring-like function of ankle extensor 333 tendons (for a review, see Alexander, 2003). Rigid segments (Fig. 7B, dashed lines), which are 334 free to rotate about joints within a plane, represent the skeleton. The ankle extensor muscles in the 335 lower leg, which originate on the femur and run along the ventral side of the tibia, are modeled as 336 a rigid strut (muscle strut, Fig. 7B, blue line) that contracts isometrically during stance in running. 337 A linear spring (tendon spring, Fig. 7B, red line), which originates from the distal end of the 338 muscle strut and extends to the digit tip, models the elastic foot tendons. The muscle strut and 339 tendon spring are ventrally offset from the midline of the skeleton at each joint by respective joint 340 radii. 341
Hind limb resilience
342 i et al. (2012),
The Journal of Experimental Biology , , 3293–3308. doi:10.1242/jeb.061937 12 Representative passive work loops (Fig. 8A C) showed that torque was higher when the foot 343 was pushed down on the solid surface than when it was extracted, similar to previous 344 observations in humans (Ker et al., 1987) and cockroaches (Dudek and Full, 2006). Maximal 345 torque was positively correlated with maximal angular displacement ( F = 64.3188, P < 0.001, 346 ANOVA). The kinks observed in the middle of the loading curve were due to the fifth toe 347 contacting the surface. Average hind limb resilience calculated from the work loops was R = 0.44 348 ± 0.12 (Fig. 8D F, 3 individuals, 64 trials). R did not differ between individuals ( F = 2.1025, P
349 = 0.1309, ANOVA), and did not depend on maximal torque ( F = 0.5208, P = 0.4732, ANOVA; 350 Fig. 8D), maximal angular displacement F = 0.0164, P = 0.8987, ANOVA; Fig. 8E , or 351 average loading rate ( F = 1.1228, P = 0.2934, ANOVA; Fig. 8F). 352 Hind foot curvature, tendon deformation, and tendon stiffness
353 The observed three-dimensional positions of the hind limb fit well to the two-dimensional hind 354 limb model (Fig. 9A D), and enabled calculation of the curvature, tendon deformation, and 355 tendon stiffness of the hind foot (see Appendix). Calculated hind foot curvature (Fig. 9E, solid 356 curve) showed that the hind foot hyperextended during stance (positive ) and flexed during 357 swing (negative ). The foot was straight at touchdown and shortly after takeoff ( ). 358 Calculated tendon spring deformation l (Fig. 9E, dashed curve) showed that the tendon spring 359 stretched during the first half and recoiled during the second half of stance. The estimated tendon 360 spring stiffness was k = 4.4 × 10 N/m (see Appendix). 361
Mechanical energetics on solid surface
362 Using the observed CoM and hind limb kinematics, calculated tendon spring stiffness and 363 deformation, and measured hind limb resilience, we examined the mechanical energetics of the 364 lizard running on the solid surface (Table 3, Fig. 9F). From the observed CoM kinematics, in the 365 first half of stance, the mechanical energy of the CoM (kinetic energy plus gravitational potential 366 energy) decreased significantly from E touchdown = 1.00 ± 0.00 at touchdown to E mid-stance = 0.81 ± 367 0.08 at mid-stance ( F = 12.2345, P = 0.0004, ANOVA, Tukey HSD). In the second half of 368 stance, the mechanical energy of the CoM recovered to E aerial = 0.95 ± 0.10 at mid aerial phase, 369 not significantly different from E touchdown (Tukey HSD). The reduction in CoM mechanical energy 370 in the first half of stance E mech = 0.19 ± 0.08 is the mechanical work needed per step on the solid 371 surface. Note that the energies of each run were normalized to E touchdown of that run. 372 i et al. (2012), The Journal of Experimental Biology , , 3293–3308. doi:10.1242/jeb.061937 13 At mid-stance, the elastic energy stored in the tendon spring was E storage = 0.18 ± 0.13 (calculated 373 from 1/2 k l max2 , see Appendix), not significantly different from E mech ( F = 0.0475, P = 374 0.8312, ANOVA). Because hind limb resilience R = 0.44 ± 0.12, the elastic recoil of the foot 375 tendons returned an energy of E return = RE storage = 0.08 ± 0.06, or 41 ± 33% of the mechanical work 376 needed per step ( E mech ) on the solid surface. We verified that foot flexion during swing induced 377 little energy storage (< 0.1 E storage ) because the hind foot was less stiff during flexion (0.7 × 10
378 N/m) than during hyperextension (4.4 × 10 N/m). 379
Granular penetration force model
380 Although little is known about the kinematics and mechanics of the complex limb intrusions 381 during legged locomotion on granular surfaces, we took inspiration from previous observations 382 that horizontal drag (Maladen et al., 2009) and vertical impact (Katsuragi and Durian, 2007) 383 forces in glass particles were insensitive to speed when intrusion speed was below approximately 384 0.5 m/s. Because the kinematics observed on the granular surface suggest that the vertical speeds 385 of most of the foot relative to the ground were below 0.5 m/s during most of the stance phase (see 386 Appendix), we assumed that the ground reaction forces on the lizard’s feet were also insensitive 387 to speed. This allowed us to use the vertical penetration force measured at 0.01 m/s to model and 388 estimate the vertical ground reaction forces on the lizard foot. 389 From the force data on both plates (Fig. 10), vertical ground reaction force F z was proportional to 390 both penetration depth |z| and projected area A of the plate (area projected into the horizontal 391 plane). F z was pointing upward during foot penetration, and pointing downward during foot 392 extraction and dropped by an order of magnitude. These measurements were in accord with 393 previous observations of forces on a sphere penetrating into granular media (Hill et al., 2005). 394 Furthermore, we estimated from free falling of particles under gravity that it would take longer 395 than the stance duration (45 ms) for the grains surrounding a penetrating foot to refill a hole 396 created by the foot of maximal depth (| z | max = 1.0 cm, see Appendix). Thus we assumed that the 397 vertical ground reaction forces were negligible during foot extraction. 398 Therefore, we approximate the vertical penetration force as: 399 F z = (cid:3420) (cid:2009)|(cid:1878)|(cid:1827), for increasing |(cid:1878)|,0, for decreasing |(cid:1878)|, (1) 400 where is the vertical stress per unit depth, which is determined by the properties of the granular 401 material and increases with compaction (Li et al., 2009). Fitting F z = | z | A to the force data 402 i et al. (2012), The Journal of Experimental Biology , , 3293–3308. doi:10.1242/jeb.061937 14 during penetration over regions where the plate was fully submerged and far from boundary (Fig. 403 10, dashed lines), we obtained = × 10 N/m for loosely packed 0.27 ± 0.04 mm diameter 404 glass particles. 405 Vertical ground reaction force on granular surface
406 During a stance on the granular surface, the CoM vertical speed v z ,CoM (calculated from z CoM ) was 407 approximately sinusoidal (Fig. 11A, dashed curve). This implies that the F z on a lizard foot must 408 be approximately sinusoidal. In addition, the foot was nearly horizontal at touchdown, but pointed 409 downward and slightly backward during takeoff. In consideration of the functional form of the 410 penetration force (Eqn. 1), we hypothesized that during stance the foot rotated subsurface by /2 411 in the sagittal plane (Fig. 11C), increasing foot depth | z | but decreasing projected foot area A , thus 412 resulting in a sinusoidal F z which reaches a maximum at mid-stance before the foot reaches 413 largest depth (see Appendix). A sinusoidal F z is also possible for a fixed projected foot area if the 414 foot maintains contact on solidified grains. However, this is unlikely considering that during 415 stance the ankle moved forward at the surface level by a foot length. 416 Assuming that during stance the hind foot rotated by /2 in the sagittal plane at a constant angular 417 velocity, the vertical ground reaction force that each foot generated was F z = 5 mg /9 sin10 t /9 T
418 (see Appendix). The net vertical acceleration due to this F z and the animal weight mg was a z = 419 F z /m – g (Fig. 11B; solid and dashed curves are a z from both hind feet, shifted from each other by 420 T /2). The CoM vertical speed v z ,CoM predicted from the total a z on both hind feet (Fig. 11A, 421 dashed curve) agreed with experimental observations (Fig. 11A, solid curve). The slight under-422 prediction of the oscillation magnitudes of v z ,CoM was likely due to an over-estimation of duty 423 factor on the granular surface. This is because F z may have dropped to zero even before takeoff if 424 the foot started moving upward before takeoff (Fig. 10). 425 Mechanical energetics on granular surface
426 Using the measured CoM kinematics, assumed foot rotation, and calculated vertical ground 427 reaction force, we examined the mechanical energetics of the lizard running on the granular 428 surface (Table 3, Fig. 11D). In the first half of stance, the mechanical energy of the CoM 429 decreased significantly from E touchdown = 1.00 ± 0.00 at touchdown to E mid-stance = 0.86 ± 0.09 at 430 mid-stance ( F = 6.6132, P = 0.007, ANOVA, Tukey HSD). In the second half of stance, the 431 mechanical energy of the CoM recovered to E aerial = 0.99 ± 0.10 at mid aerial phase, not 432 significantly different from E touchdown (Tukey HSD). The reduction in CoM mechanical energy in 433 i et al. (2012), The Journal of Experimental Biology , , 3293–3308. doi:10.1242/jeb.061937 15 the first half of stance E mech = 0.14 ± 0.09 is the mechanical work needed per step on the 434 granular surface. By integration of F z over vertical displacement of the foot during stance (see 435 Appendix), the energy lost to the granular substrate per step was estimated as E substrate = 0.17 ± 436 0.05, not significantly different from E mech ( F = 0.4659, P = 0.5078, ANOVA). Note that the 437 energies of each run were normalized to E touchdown of that run. 438 Discussion
Conservation of spring-mass-like CoM dynamics on solid and granular surfaces
441 The observed kinematics and calculated mechanical energetics demonstrated that the zebra-tailed 442 lizard ran like a spring-mass system on both solid and granular surfaces. On both surfaces, the 443 CoM forward speed (Fig. 5B), vertical position (Fig. 5C), and lateral position (Fig. 5D) displayed 444 oscillation patterns that are in accord with predictions from the Spring-Loaded Inverted Pendulum 445 (SLIP) model (Blickhan, 1989) and the Lateral Leg Spring (LLS) model (Schmitt et al., 2002). 446 The small relative oscillations of the CoM forward speed (i.e., v x ,CoM / v x ,CoM << 1) was expected 447 because the Froude number was large for the lizard (see Appendix). The substantial sprawling of 448 the legs contributed to the medio-lateral oscillatory motion of the animal. Furthermore, on both 449 surfaces, the mechanical energy of the CoM oscillated within a step, reaching minimum at mid-450 stance and maximum during the aerial phase (Fig. 9F, 11D), a defining feature of spring-mass 451 like running (Blickhan, 1989; Schmitt et al., 2002). 452 To our knowledge, ours is the first study to quantitatively demonstrate spring-mass-like CoM 453 motion in lizards running on granular surfaces. Spring-mass-like CoM motion was previously 454 observed in other lizards and geckos running on solid surfaces (Farley and Ko, 1997; Chen et al., 455 2006), but it was not clear whether energy-saving by elastic elements played an important role. 456 Hind foot function on solid surface: energy-saving spring
457 Our study is also the first to quantify elastic energy savings in foot tendons in lizards during 458 running on solid surfaces. The significant energy savings (about 40% of the mechanical work 459 needed per step) in the zebra-tailed lizard’s hind foot tendons is in a similar range to the energy 460 savings by ankle extensor tendons and digital flexor tendons and ligaments in larger animals 461 (Alexander, 2003), such as kangaroos (50%, Alexander and Vernon, 1975), wallabies (45%, 462 i et al. (2012),
The Journal of Experimental Biology , , 3293–3308. doi:10.1242/jeb.061937 16 Biewener et al., 1998), horses (40%, Biewener, 1998b), and humans (35%, with an additional 17% 463 from ligaments in the foot arch, Ker et al., 1987). 464 This is surprising considering that the elastic energy saving mechanism was previously thought 465 less important in small animals (e.g., 14% in hopping kangaroo rat of ~ 100 g mass, Biewener et 466 al., 1981). Because the tendons of small animals are “overbuilt” to withstand large stresses during 467 escape, during steady-speed locomotion these tendons usually experience stresses too small to 468 induce significant elastic energy storage and return (Biewener and Blickhan, 1988; McGowan et 469 al., 2008). We verified that for zebra-tailed lizards running at ~ 1 m/s the maximal stress in the 470 foot tendons is 4.3 MPa (see Appendix), well below the 100 MPa breaking stress for most 471 tendons (Kirkendall and Garrett, 1997). 472 The zebra-tailed lizard’s elongate hind foot and digitigrade foot posture on the solid surface may 473 be an adaptation for elastic energy saving during rapid locomotion. Like other iguanids (Russell, 474 1993), this lizard does not have substantial ankle extensor tendons as large animals do. 475 Nevertheless, elongation of foot tendons and a digitigrade posture enhance the hind foot’s energy 476 saving capacity by decreasing tendon stiffness and mechanical advantage (Biewener et al., 2004) 477 (see Appendix). A recent study also found significant energy savings (53%) by elongate foot 478 tendons in running ostriches (Rubenson et al., 2011). More generally, elongation of distal limb 479 segments such as legs, feet, and toes which possess tendons may be an adaptation for energy 480 saving during rapid locomotion. Indeed, many cursorial animals including mammals (Garland Jr. 481 and Janis, 1993), lizards (Bauwens et al., 1995), and dinosaurs (Coombs Jr., 1978) display 482 elongation of distal limb segments. Short fascicles and long tendons and ligaments are often 483 found in the ankle extensor muscles and digital flexor muscles in large cursorial ungulates such as 484 horses, camels, and antelopes (Alexander, 2003). 485 Solid surface model assumptions
486 Our estimates of elastic energy storage and return on the solid surface assume isometric 487 contraction of lower leg muscles. However, muscles have a finite stiffness and do lengthen by a 488 small amount under limb tension (Biewener, 1998a; Roberts et al., 1997). Despite this difference, 489 our estimates still hold, because in the latter case both lower leg muscles and foot tendons behave 490 like springs, and the total stiffness remains the same (since external force and total deformation 491 remain the same). In the case where the muscles actively shorten during stance and further 492 lengthen the tendons (which does positive mechanical work on the tendons), the energy storage 493 and return in the tendons would increase. However, the overall energy efficiency would decrease 494 i et al. (2012),
The Journal of Experimental Biology , , 3293–3308. doi:10.1242/jeb.061937 17 (with everything else being the same), because apart from energy lost in tendon recoil, energy is 495 further lost in the muscles that perform the mechanical work, i.e., muscle work is more expensive 496 than tendon work (Biewener and Roberts, 2000). 497 In addition, the hind limb resilience obtained from anesthetized lizards was assumed to be a good 498 estimate for hind limb resilience in running lizards. This is based on our observations that 499 resilience was independent of torque, angular displacement, and loading rate, as well as previous 500 findings that the damping properties of animal limbs are largely intrinsic to their structure and 501 material properties (Weiss et al., 1988; Fung, 1993; Dudek and Full, 2006). Future studies using 502 techniques such as tendon buckles (Biewener et al., 1998), sonomicrometry (Biewener et al., 503 1998), ultrasonography (Maganaris and Paul, 1999), and oxygen consumption measurement 504 (Alexander, 2003) during locomotion are needed to confirm this assumption. 505 Hind foot function on granular surface: dissipative, force-generating paddle
506 The similarity between the observed and predicted v z ,CoM on the granular surface supports the 507 hypothesis of subsurface foot rotation. We speculate that on the granular surface the foot 508 functions as a “paddle” through fluidized grains to generate force. This differs from previous 509 observations of the utilization of solidification forces of the granular media in a legged robot (Li 510 et al., 2009; Li et al., 2010b) and sea turtle hatchlings (Mazouchova et al., 2010) moving on 511 granular surfaces. 512 As the zebra-tailed lizard’s hind foot paddles through fluidized grains to generate force, energy is 513 lost to the substrate because grain contact forces in granular media are dissipative (Nedderman, 514 1992). A large foot can reduce energy loss to the granular substrate compared to a small one, 515 much like large snowshoes used by humans can reduce energy cost for walking on snow (Knapik 516 et al., 1997). From our model of foot-ground interaction on the granular surface, for a given 517 animal (constant weight), energy loss to the substrate is proportional to foot penetration depth, 518 and thus inversely proportional to foot area and substrate strength (see Appendix). 519 Granular surface model assumptions
520 In our modeling of the foot-ground interaction on the granular surface using the penetration force 521 model, we made two assumptions. First, we assumed that the ground reaction forces were 522 insensitive to speed. This is true in the low speed regime (<0.5 m/s for our glass particles, 523 Maladen et al., 2009) where particle inertia is negligible and forces are dominated by particle 524 friction. Because friction is proportional to pressure, and pressure is proportional to depth (Hill et 525 i et al. (2012),
The Journal of Experimental Biology , , 3293–3308. doi:10.1242/jeb.061937 18 al., 2005), granular forces in the low-speed regime are proportional to depth ( F z = | z | A ), 526 analogous to the hydrostatic forces in fluids ( F z = g | z | A , i.e. buoyant forces due to hydrostatic 527 pressure). 528 Second, we used the vertical stress per unit depth determined from vertical penetration of a 529 horizontally oriented disc to estimate forces on the foot as it rotates subsurface. In this calculation, 530 the effective vertical stress per unit depth cos foot (see Appendix) depended on foot orientation 531 via a simple relation cos foot (because projected area A=A foot cos foot ; see Appendix), and not on 532 direction of motion. However, our recent physics experiments (Li et al., in preparation) suggest 533 that stresses in granular media in the low speed regime depend on both orientation and direction 534 of motion in a more complicated manner, and that cos foot overestimates vertical stress per unit 535 depth for all foot orientations and directions of motion except when the foot is horizontal and 536 moving vertically downwards. Therefore, our model must be overestimating hydrostatic-like 537 forces, and there must be additional forces contributing to the lizard’s ground reaction forces. 538 We propose that these additional forces are likely from hydrodynamic-like inertial forces 539 resulting from the local acceleration of the substrate (particles) by the foot. Analogous to 540 hydrodynamic forces in fluids (Vogel, 1996), for an intruder moving rapidly in granular media, 541 the particles initially at rest in front of the intruder are accelerated by, and thus exert reaction 542 forces on, the intruder. Hydrodynamic-like forces at ~1 m/s can play an important role both in 543 impact forces on free falling intruders (Katsuragi and Durian, 2007; Goldman and Umbanhowar, 544 2008) and in legged locomotion of small lightweight robots (Qian et al., 2012). We note that the 545 foot rotation hypothesis should hold regardless, because hydrodynamic-like forces are also 546 proportional to projected area (Katsuragi and Durian, 2007). 547 However, we know too little about the lizard’s subsurface foot kinematics and the force laws in 548 the high-speed regime on an intruder being pushed in a complex path within granular media (not 549 simply a free-falling intruder) to more accurately calculate both hydrostatic-like and 550 hydrodynamic-like forces. Future x-ray high-speed imaging experiments (Maladen et al 2009; 551 Sharpe et al., 2012) will reveal how the lizard foot was moving subsurface. Further studies of 552 intrusion forces in granular media in both low-speed (Li et al., 2013) and high-speed regimes can 553 provide a more comprehensive understanding of ground reaction forces during legged locomotion 554 on granular surfaces and provide better estimates of foot penetration depth and energy loss. 555 Comparison to water-running in basilisk lizard
556 i et al. (2012),
The Journal of Experimental Biology , , 3293–3308. doi:10.1242/jeb.061937 19 The rapid impact of the foot on the surface at touchdown and hypothesized subsurface foot 557 rotation appear kinematically similar to the slap and stroke phases of the basilisk lizards running 558 on the surface of water (Glasheen and McMahon, 1996a; Hsieh, 2003). For the zebra-tailed lizard 559 running on sand, both granular hydrostatic-like and hydrodynamic-like forces can contribute to 560 vertical ground reaction force. This is also qualitatively similar to water-running basilisk lizard, 561 which utilizes both hydrostatic forces resulting from the hydrostatic pressure between the water 562 surface and the bottom of the air cavity created by the foot, and hydrodynamic forces resulting 563 from the water being accelerated from rest by the rapidly moving foot (Glasheen and McMahon, 564 1996a, 1996b; Hsieh and Lauder, 2004). 565 However, the degree to which each species relies on these two categories of forces differs due to 566 differences in the properties of the supporting media. For given foot size, depth, and speed, the 567 hydrostatic(-like) forces in water are an order of magnitude smaller than the hydrostatic-like 568 forces in granular media, whereas the hydrodynamic(-like) forces are similar between in water 569 and in granular media (see Appendix). As a result, the basilisk lizard running on water must rely 570 on hydrodynamic forces to a larger degree than the zebra-tailed lizard running on sand, 571 considering that these two lizards have similar size (~ 0.1 m). An extreme example for this is that 572 it is impossible for a basilisk lizard to stand on the surface of water, but a zebra-tailed lizard can 573 stand on loose sand. 574 Motor function of upper hind leg
575 Despite the passive nature of the leg spring in the spring-mass model, animal limbs do not 576 function purely passively as springs—the muscles within them must perform mechanical work. 577 We have shown that on the solid surface, the lizard’s hind foot saves about 40% of the 578 mechanical work per step. The remaining 60% is lost either within the foot or to the ground, and 579 must be compensated by mechanical work performed by muscles, which is W muscle = 0.11 ± 0.10. 580 This work is likely provided by knee extension during the second half of stance (Fig. 6D, red 581 curve) powered by the upper leg muscles. 582 On the granular surface, substantial energy is lost to the substrate. This is in accord with previous 583 observations of higher mechanical energetic cost during locomotion on granular surfaces in 584 human (Zamparo et al., 1992; Lejeune et al., 1998) and legged robots (Li et al., 2010a). Because 585 the energy lost to the substrate equals the reduction in CoM mechanical energy during the first 586 half of stance, even without energy loss within the limb, the upper hind leg muscles must perform 587 mechanical work of W muscle = 0.31 ± 0.10 during the second half of stance, about three times that 588 i et al. (2012), The Journal of Experimental Biology , , 3293–3308. doi:10.1242/jeb.061937 20 on the solid surface for a given animal running at a given speed, as evidenced by the larger knee 589 extension on the granular surface (Fig. 6D, blue curve). 590 Our models of the foot-ground interaction on both surfaces assume purely passive foot mechanics, 591 and do not consider the role of active neurosensory control. However, animals can actively adjust 592 kinematics and muscle function to accommodate changes in surface conditions (Ferris et al., 1999; 593 Daley and Biewener, 2006). We observed that when confronted by a substrate which transitioned 594 from solid into granular (or vice versa ), the lizard displayed partial adjustment of foot posture 595 during the first step on the new surface, followed by full adjustment during the second step. 596 Future studies using neuromechanics techniques, such as EMG (Biewener et al., 1998; Sponberg 597 and Full, 2008; Sharpe and Goldman, in review) and denervation and reinnervation (Chang et al., 598 2009), can determine how neural control and sensory feedback mechanisms are used to control 599 limb function to accommodate changing substrates. 600 Conclusions
601 During running on both solid and granular surfaces, the zebra-tailed lizard displayed spring-mass-602 like center of mass kinematics with distinct hind foot, hind leg, and trunk kinematics. The lizard’s 603 large, elongate hind foot served multiple functions during locomotion. On the solid surface, the 604 hind foot functioned as an energy-saving spring and reduced about 40% of the mechanical work 605 needed each footstep. On the granular surface, the hind foot paddled through fluidized grains to 606 generate force, and substantial energy was lost during irreversible deformation of the granular 607 substrate. The energy lost within the foot and to the substrate must be compensated for by 608 mechanical work done by the upper hind leg muscles. 609 The multifunctional hind foot may passively (and possibly actively) adjust to the substrate during 610 locomotion in natural terrain, and provide this desert generalist with energetic advantages and 611 simplify its neurosensory control tasks (Full and Koditschek, 1999). Current robotic devices often 612 suffer performance loss and high cost of transport on flowing substrates like granular material 613 (Kumagai 2004; Li et al., 2009; Li et al., 2010b; Li et al., 2010a). Insights from studies like ours 614 can provide inspiration for next-generation multi-terrain robots (Pfeifer et al., 2007). Finally, our 615 study also highlights the need for comprehensive force models for granular media (Li et al., in 616 preparation) and for flowing terrestrial environments in general. 617
Appendix
619 i et al. (2012),
The Journal of Experimental Biology , , 3293–3308. doi:10.1242/jeb.061937 21 Small relative oscillation in forward speed
620 Running at 1.1 m/s, the lizard’s Froude number in the sagittal plane was Fr = v x ,CoM2 / gL = 3 621 (where L ≈
4 cm is the leg length at touchdown), above the typical value of 2.5 where most 622 animals transition from trotting to galloping (Alexander, 2003). This implied that the kinetic 623 energy (½ mv CoM2 ≈ ½ mv x ,CoM2 ) of the CoM was 3 times larger than its gravitational potential 624 energy ( mgz CoM ). Because both the forward speed oscillation v x ,CoM and vertical speed oscillation 625 v z ,CoM were determined by the total ground reaction force and the attack angle of the leg spring 626 ( = sin -1 ( v x ,CoM DT /2 L ) = 0.9 rad), they must be of the same order of magnitude (Blickhan, 1989), 627 i.e., v x ,CoM ~ v z ,CoM . From the observed CoM kinematics, v z ,CoM < ( gL ) . Therefore, v x ,CoM ~ 628 v z ,CoM < ( gL ) << v x ,CoM , and v x ,CoM / v x ,CoM << 1. 629 Hind foot curvature on solid surface
630 Three-dimensional kinematics showed that the hind limb (from the hip to the digit tip of the 631 fourth toe) remained nearly within a plane during the entire stride (out-of-plane component is 5% 632 averaged over the entire stride). During stance, the orientation of the foot plane remained nearly 633 unchanged, with a foot sprawl angle of 53 ± 4° relative to the sagittal plane in the posterior view. 634 Hind foot curvature could then be obtained by fitting a circle to the hind foot (from the ankle to 635 the digit tip) within the foot plane and determining the radius of curvature of the fit circle (see 636 diagram in Fig. 9A), i.e., = ± 1/ , where + sign indicates foot hyperextension, sign indicates 637 foot flexion, and = 0 indicates a straight foot. 638 Tendon spring deformation
639 From the two-dimensional strut-spring model of the hind limb, by geometry, the tendon spring 640 deformation l was related to the observed changes of joint angles and the foot joint radii as: l = i r i i , where i = K, A, MP, PP were the four joints in the model i the observed changes of 642 joint angles, and r i the joint radii ( r K = r A = 1.25 mm, r MP = 0.75 mm, r PP = 0.50 mm). We 643 observed that the relaxed hind foot of a live animal was nearly straight (Fig. 1A), which was 644 similar to the foot shape at touchdown during running (Fig. 3A,E). Thus we defined the relaxed 645 length of the tendon spring as the length when the foot was straight, i.e., l = 0 at touchdown. 646 Calculated maximal tendon spring deformation l max = 0.78 mm corresponded to a 3% strain. We 647 did not consider tendon spring deformation in the swing phase (dotted curve in Fig. 6F) because 648 the assumption of isometric contraction of lower leg muscles was only valid for the stance phase. 649 i et al. (2012), The Journal of Experimental Biology , , 3293–3308. doi:10.1242/jeb.061937 22 Tendon spring stiffness
650 The stiffness of the tendon spring was defined as the maximal tension divided by the maximal 651 deformation of the tendon spring, i.e., k = T max / l max . From the observed CoM kinematics, the 652 total ground reaction force at mid-stance was F max = 0.3 N within the coronal plane and pointed 653 from the digit tip to the hip. At mid-stance, because the foot was neither dorsiflexing nor 654 plantarflexing, torque was balanced at the ankle, i.e., T max r A = F max x AT , where x AT = 1.4 cm was 655 the horizontal distance between the ankle and the digit tip at mid-stance, and r A = 1.25 mm. Thus 656 T max = 3.4 N and k = 4.4 × 10 N/m. The maximal stress in the foot tendons during stance was 657 max = T max / r PP2 = 4.3 MPa. 658 The torsional stiffness of the ankle observed in anesthetized lizards from the modified work loop 659 experiments (~ 1 × 10 Nm/rad) was an order of magnitude smaller than estimated from running 660 kinematics (12 × 10 Nm/rad). This is however not contradictory but expected because during 661 stance the lizard’s lower leg muscles must be activated, and the resulting higher tension from 662 muscle contraction increases limb stiffness (Weiss et al., 1988). 663
Foot elongation increases energy savings on solid surface
664 The stiffness of a piece of elastic material like a tendon is k = E A /l , where E is the Young’s 665 modulus, A the cross sectional area, and l the rest length of the material. Most animal tendons 666 are primarily made of collagen (Kirkendall and Garrett, 1997) and are of similar Young’s 667 modulus (i.e., E is nearly constant). Thus, the stiffness of the tendon spring scales as k ∝ A /l ∝ r /l , i.e., an elongate tendon (smaller r and larger l ) is less stiff and stretches more easily than 669 a short, thick tendon. Because elastic energy storage decreases with tendon stiffness ( E storage = ½ 670 k l max2 = ½ T max2 / k ∝ k for a given T max ), an elongate tendon can store (and return) more energy. 671 An elongate foot also reduces the moment arm of tendon tension (small r A ) but increases the 672 moment arm of the ground reaction force (large x AT ) about the ankle, therefore reducing the 673 mechanical advantage (Biewener et al., 2004), so it increases tension in the foot for a given 674 ground reaction force (because T max = F max x AT / r A ) and amplifies tendon stretch for enhanced 675 energy storage and return. 676 Vertical ground reaction force on granular surface
677 We assumed that the hind foot was rotating at a constant angular velocity about the moving 678 ankle during stance, i.e., foot = t within 0 ≤ t ≤ DT and 0 ≤ foot ≤ /2, then DT = 10 /9 T
679 i et al. (2012),
The Journal of Experimental Biology , , 3293–3308. doi:10.1242/jeb.061937 23 = 35 rad/s. From the measured vertical speed of the ankle and this assumed foot rotation, the 680 vertical speed of most (75%) of the foot was always below 0.5 m/s during most (75%) of stance.
681 Given foot rotation foot = t , the foot area projected in the horizontal plane decreased with time 682 as A = A foot cos t , where A foot = 1 cm is the hind foot area; the foot depth (measured at the center 683 of the foot) increased with time as | z| = | z| max sin t . The vertical ground reaction force on the foot 684 was then sinusoidal: F z = F z ,max sin2 t , which was sinusoidal, where F z ,max = 685 A foot |z| max sin /4cos /4 = ½ A foot |z| max . For steady state locomotion on a level surface, the F z
686 generated by one foot averaged over a cycle must equal half the body weight, i.e., 687 (cid:1516) (cid:1832) (cid:3053),(cid:3040)(cid:3028)(cid:3051) (cid:1871)(cid:1861)(cid:1866)2(cid:2033)(cid:1872) (cid:1856)(cid:1872) (cid:3021)(cid:2868) = ½ mg . Therefore, F z ,max = 5 mg /9 and F z = 5 mg /9 sin10 t/9 T . 688 Energy loss to granular substrate
689 By integration of vertical ground reaction force over vertical displacement of the foot, the energy 690 loss to the granular substrate was E substrate = (cid:1516) (cid:1832) (cid:3053)|(cid:3053)| (cid:3288)(cid:3276)(cid:3299) (cid:2868) (cid:1856)|(cid:1878)| = (cid:1516) (cid:1832) (cid:3053)(cid:3021)(cid:2868) (cid:3031)|(cid:3053)|(cid:3031)(cid:3047) (cid:1856)(cid:1872) , where | z| max = 1.0 cm 691 from F z ,max = ½ A foot |z| max . The hypothesized foot rotation in the sagittal plane did not take into 692 account the sprawl of the foot during stance, which could induce additional energy loss by lateral 693 displacement of the granular substrate. However, a sprawled foot posture did not affect the 694 condition of vertical force balance and thus did not change our estimate of energy dissipation in 695 the sagittal plane. Therefore this estimate provides a lower bound. 696 Large foot area reduces energy loss on granular surface
697 For a given animal (constant weight mg ), F z ,max = ½ A foot |z| max = 5 mg /9 is constant, thus E substrate
698 = |(cid:1878)| (cid:3040)(cid:3028)(cid:3051) (cid:1516) (cid:1832) (cid:3053)(cid:3021)(cid:2868) (cid:2033) (cid:1855)(cid:1867)(cid:1871)(cid:2033)(cid:1872) (cid:1856)(cid:1872) ∝ | z | max ∝ A foot ). This implies that the energy loss to the granular 699 substrate increases with foot penetration depth. On a given granular surface (fixed ), a larger 700 foot (larger A foot ) sinks less than a smaller foot, and thus loses less energy to the substrate. For a 701 given foot size (fixed A foot ), a foot sinks less on a stronger granular substrate (larger ) than on a 702 weaker substrate, and thus loses less energy to the substrate. 703 Comparison of forces in granular media and in water
704 For water, hydrostatic force is F z = g | z | A . Comparing this with F z = | z | A for granular media, g
705 is the equivalent of . For water, g = 1.0 × 10 N/m ; for loosely packed glass particles N/m . Therefore, the hydrostatic forces in water are an order of magnitude smaller than the 707 hydrostatic-like forces in granular media for given foot size and depth. 708 i et al. (2012), The Journal of Experimental Biology , , 3293–3308. doi:10.1242/jeb.061937 24 Hydrodynamic(-like) forces should be proportional to the density of the surrounding media 709 because they are due to the media being accelerated. For water, = 1.0 × 10 N/m ; for loosely 710 packed glass particles the effective density is 2.5 × 10 N/m × 0.58 volume fraction = 1.45 × 10
711 N/m . Therefore, the hydrodynamic forces in water and hydrodynamic-like forces in granular 712 media are on the same order of magnitude for given foot size and foot speed. 713 Acknowledgements
715 We gratefully thank Sarah Sharpe, Yang Ding, Nick Gravish, Ryan Maladen, Paul Umbanhowar, 716 Kyle Mara, Young-Hui Chang, Andy Biewener, Tom Roberts, Craig McGowan, and two 717 anonymous reviewers for helpful discussions and/or comments on the manuscript; Loretta Lau for 718 help with kinematics data tracking; Sarah Sharpe for help with animal protocol and 719 anesthetization; Mateo Garcia, Nick Gravish, and Andrei Savu for help with force plate setup; 720 Ryan Maladen and The Sweeney Granite Mountains Desert Research Center for help with animal 721 collection; and the staff of The Physiological Research Laboratory animal facility of The Georgia 722 Institute of Technology for animal housing and care. This work was funded by The Burroughs 723 Wellcome Fund (D.I.G. and C.L.), The Army Research Laboratory Micro Autonomous Systems 724 and Technology Collaborative Technology Alliance (D.I.G. and C.L.), The Army Research 725 Office Biological Locomotion Principles and Rheological Interaction Physics (D.I.G. and C.L.) 726 and The University of Florida and Temple University start-up funds (S.T.H.). 727
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Physiology , 183–187. 906 907 908 i et al. (2012), The Journal of Experimental Biology , , 3293–3308. doi:10.1242/jeb.061937 32 909 Fig. 1. Model organism and three-dimensional kinematics experiments. (A) A zebra-tailed lizard 910 resting on sand in the wild (photo: Thomas C. Brennan). (B) Experimental setup for three-911 dimensional kinematics capture, with definitions of pelvis height ( z pelvis ), knee height ( z knee ), trunk 912 pitch angle ( pitch ), and knee angle ( knee ). Colored dots in (A,B) are digitized points on the 913 midline of the trunk, hind leg, and elongate hind foot. 914 i et al. (2012), The Journal of Experimental Biology , , 3293–3308. doi:10.1242/jeb.061937 33 915 Fig. 2. Experiments to measure hind limb resilience and granular penetration force. (A) 916 Experimental setup for hind limb resilience measurements. Dashed foot tracing shows the relaxed, 917 straight foot right before touchdown. Solid foot tracing shows the hyperextended foot during 918 ground contact. F , ground reaction force; , angle between the ankle and the digit tip in the 919 relaxed, straight foot; t , angle between the ankle and the digit tip in the hyperextended foot; , 920 torque about the ankle. (B) Experimental setup for granular penetration force measurements. 921 i et al. (2012), The Journal of Experimental Biology , , 3293–3308. doi:10.1242/jeb.061937 34 922 Fig. 3. Lateral views of representative runs on the solid (A D) and the granular (F I) surface (see 923 Movies 1, 2 in supplementary material). (E,J) Closer views of foot posture at touchdown showing 924 definition of touchdown foot angle touchdown . Solid lines and curves along the foot indicate hind 925 foot posture and shape. Note that the lateral camera was oriented at an angle to the x, y, z axes 926 such that forward (+ x ) direction appeared to point slightly downwards. 927 i et al. (2012), The Journal of Experimental Biology , , 3293–3308. doi:10.1242/jeb.061937 35 928 Fig. 4. Performance and gait on the solid (red) and the granular (blue) surfaces. (A) Average 929 forward speed vs. stride frequency. (B) Duty factor vs. average forward speed. Different symbols 930 represent different individuals. Filled symbols are from the seven representative runs for each of 931 the seven individuals tested on both substrates. Empty symbols are from runs that were not 932 included in the representative data set. 933 i et al. (2012), The Journal of Experimental Biology , , 3293–3308. doi:10.1242/jeb.061937 36 934 Fig. 5. Center of mass (CoM) kinematics (mean ± s.d.) vs. time during a stride on the solid (red) 935 and the granular (blue) surfaces. (A) Footstep pattern. (B) CoM forward speed. (C) CoM vertical 936 position. (D) CoM lateral position. See Fig. 1 for definitions of kinematic variables. 937 i et al. (2012), The Journal of Experimental Biology , , 3293–3308. doi:10.1242/jeb.061937 37 938 Fig. 6. Hind foot, hind leg, and trunk kinematics (mean ± s.d.) vs. time during a stride on the solid 939 (red) and the granular (blue) surfaces. (A) Touchdown foot angle. (B) Knee height. (C) Pelvis 940 height. (D) Knee angle. (E) Trunk pitch angle. See Fig. 1 and Fig. 3E,J for definitions of 941 kinematic variables. 942 i et al. (2012), The Journal of Experimental Biology , , 3293–3308. doi:10.1242/jeb.061937 38 943 Fig. 7. Anatomy and a strut-spring model of the hind limb. (A) Ventral anatomy of a dissected 944 hind limb. Lower hind leg muscles are marked in blue; foot tendons are marked in red. (B) A 945 two-dimensional model of the hind limb. The muscle strut models isometrically contracting lower 946 leg muscles; the tendon spring models foot tendons. The radii of colored circles correspond to 947 measured joint radii. 948 i et al. (2012), The Journal of Experimental Biology , , 3293–3308. doi:10.1242/jeb.061937 39 949 950 Fig. 8. Hind limb resilience. (A C) Representative passive work loops of the hind foot (measured 951 at the digit tip) from each of the three anesthetized lizards tested. Different curves are from 952 different trials. The area within a work loop is the energy lost within the foot. See Fig. 2A for 953 schematic of experimental setup. (D F) Hind limb resilience vs. maximal torque, maximal 954 angular displacement, and average loading rate. Different symbols are from different individuals. 955 Solid and dashed lines in (D F) denote mean ± s.d. 956 957 i et al. (2012),
The Journal of Experimental Biology , , 3293–3308. doi:10.1242/jeb.061937 40 958 Fig. 9. Foot-ground interaction on the solid surface. (A D) The hind foot shape from the lateral 959 view of a representative run on the solid surface. (A D) correspond with (A D) in Fig. 3. The 960 hind foot shape in the dorsal view is similar because the sprawl angle of the foot plane is nearly 961 constant during stance. The diagram in (B) defines the radius of curvature of the foot (see 962 Appendix). (E) Foot curvature (solid) and tendon spring deformation (dashed) (mean ± s.d.) vs. 963 time during a stride on the solid surface. Tendon spring deformation is not meaningful during 964 swing (dotted) when the muscle strut assumption does not hold. (F) Mechanical energies of the 965 CoM and elastic energies of the foot (mean ± s.d.) on the solid surface. All energies are 966 normalized to the mechanical energy of the CoM at touchdown ( E touchdown ) for each run. * 967 indicates that E mid-stance is significantly different from E touchdown and E aerial ( P < 0.05, ANOVA, 968 Tukey HSD).
969 i et al. (2012),
The Journal of Experimental Biology , , 3293–3308. doi:10.1242/jeb.061937 41
970 Fig. 10. Granular penetration force (mean ± s.d.) vs. depth on two plates of different areas: A = 971 7.6 × 2.5 cm and A = 3.8 × 2.5 cm . See Fig. 2B for schematic of experimental setup. Dashed 972 lines are linear fits to the data over steady state during penetration using Eqn. (1). 973 974 i et al. (2012), The Journal of Experimental Biology , , 3293–3308. doi:10.1242/jeb.061937 42 975 Fig. 11. Foot-ground interaction on the granular surface. (A) CoM vertical speed (mean ± s.d.) vs. 976 time during a stride. Solid curve is from experiment. Dashed curve is calculated from the vertical 977 acceleration from the model. (B) Vertical acceleration vs. time during a stride calculated from the 978 total vertical ground reaction force F z on both feet and the animal weight mg . Solid and dashed 979 curves are the F z on the two alternating hind feet. (C) Hypothesized subsurface foot rotation in the 980 sagittal plane. (F I) correspond with (F I) in Fig. 3. foot , foot angle in the vertical plane. (D) 981 Mechanical energy of the CoM and the energy loss to the substrate (mean ± s.d.) during running 982 on the granular surface. All energies are normalized to the mechanical energy of the CoM at 983 touchdown ( E touchdown ) for each run. * indicates that E mid-stance is significantly different from 984 E touchdown and E aerial ( P < 0.05, ANOVA, Tukey HSD). 985 986 i et al. (2012), The Journal of Experimental Biology , , 3293–3308. doi:10.1242/jeb.061937 43 Table 1. Morphological measurements (mean ± s.d.) of the seven individuals tested in the 3-D 987 kinematics experiments. 988 SVL (cm) 7.2 ± 0.6 Mass m (g) 11.0 ± 2.7 Trunk length (cm) 4.4 ± 0.4 Pelvic width (cm) 1.4 ± 0.1 Hind limb length (cm) 6.4 ± 0.1 Hind foot length (cm) 2.7 ± 0.1 Femur length (cm) 1.6 ± 0.2 Tibia length (cm) 2.1 ± 0.2 Tarsals and metatarsals length (cm) 1.0 ± 0.1 Fourth toe length (cm) 1.7 ± 0.1 989 i et al. (2012), The Journal of Experimental Biology , , 3293–3308. doi:10.1242/jeb.061937 44 Table 2. Gait and kinematic variables (mean ± s.d.) and statistics using an ANCOVA. P values 990 reported are for substrate effect. 991 Variable Solid Granular F P †Average forward speed v (cid:3364) x ,CoM (m/s) 1.2 ± 0.3 1.1 ± 0.3 0.4784 0.5023 Stride frequency f (Hz) 7.5 ± 1.6 8.1 ± 2.0 9.9101 Duty factor D (m) 0.16 ± 0.02 0.14 ± 0.02 8.9112 Average CoM height (cid:1878)̅
CoM (cm) 3.2 ± 0.7 2.2 ± 0.5 5.4690
Magnitude of CoM vertical oscillations z CoM (cm) 0.3 ± 0.2 0.4 ± 0.3 3.7031 0.4697 Lowest CoM height z CoM (cm) 3.0 ± 0.7 2.0 ± 0.4 7.7544
Time of lowest CoM height ( T ) 0.18 ± 0.04 0.19 ± 0.04 0.9696 0.6366 Highest CoM height z CoM (cm) 3.3 ± 0.7 2.4 ± 0.6 3.6126
Time of highest CoM height ( T ) 0.44 ± 0.04 0.48 ± 0.01 3.0642 Magnitude of CoM lateral oscillations y CoM (cm) 0.86 ± 0.19 0.94 ± 0.23 0.2350 0.5263 Average pelvis height pelvis (cid:1878)̅ pelvis (cm) 3.1 ± 0.7 1.9 ± 0.5 8.8912
Average trunk pitch angle (cid:3364) pitch (deg) 1 ± 3 9 ± 2 19.5282 Touchdown knee height z knee (cm) 2.7 ± 0.7 1.7 ± 0.6 6.7157 Lowest knee height z knee (cm) 1.8 ± 0.5 0.7 ± 0.4 15.4261 Knee vertical displacement during stance z knee (cm) 0.9 ± 0.2 1.1 ± 0.4 0.7128 0.3056 Touchdown knee angle knee (deg) 88 ± 25 90 ± 13 1.2344 0.6713 Lowest knee angle knee (deg) 79 ± 17 79 ± 10 1.3175 0.7549 Highest knee angle knee (deg) 116 ± 15 150 ± 8 17.568 Knee joint extension during stance knee (deg) 37 ± 13 71 ± 4 18.0994 ‡Average leg sprawl angle during stance sprawl (deg) 40 ± 1 38 ± 5 N/A N/A Touchdown foot angle touchdown (deg) 12 ± 4 4 ± 3 7.6973 All significant differences ( P < 0.05) are in bold. Degree of freedom is (2,11) for all variables. 992 † An ANOVA was used to test the effect of substrate on running speed. 993 ‡ A direct comparison was not possible for sprawl between substrates because sprawl was measured 994 differently: on the solid surface, leg orientation was measured from the hip to the digit tip; on the 995 granular surface, leg orientation was measured from the hip to the ankle. 996 997 i et al. (2012), The Journal of Experimental Biology , , 3293–3308. doi:10.1242/jeb.061937 45 Table 3. Normalized energetic variables (mean ± s.d.). All energies were normalized to E touchdown