Dynamical determinants of different spine movements and gait speeds in rotary and transverse gallops

Quadruped gallop is categorized into two types: rotary and transverse. While the rotary gallop involves two types of flight with different spine movements, the transverse gallop involves only one type of flight. The rotary gallop can achieve faster locomotion than the transverse gallop. To clarify these mechanisms from a dynamic viewpoint, we developed a simple model and derived periodic solutions by focusing on cheetahs and horses. The solutions gave a criterion to determine the flight type: while the ground reaction force does not change the direction of the spine movement for the rotary gallop, it changes for the transverse gallop, which was verified with the help of animal data. Furthermore, the criterion provided the mechanism by which the rotary gallop achieves higher-speed than the transverse gallop based on the flight duration. These findings improve our understanding of the mechanisms underlying different gaits that animals use.


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Quadruped animals use different gaits depending on locomotion speed, walking at low speeds 24 and changing their gait to a trot and a canter to increase speed. In the highest range of the speed, 25 the gait changes to a gallop. The galloping gait is generally categorized into two types: rotary 26 and transverse gallop, involving different footfall sequences ( Fig. 1) (Hildebrand, 1977). The  verse gallops. However, the mechanisms producing the different flight phases between the rotary 41 and transverse galloping gaits remain unclear. 42 In galloping gait, animals flex and extend their spine during flight phases, as observed in chee- 43 tahs. Although horses exhibit only a small amount of spine bending (Gyambaryan, 1974), there is 44 some motion at the sacrum (Hildebrand, 1959 improves energy efficiency because the energy is stored in the elastic elements of the body then 48 released (Taylor, 1978;Alexander, 1988; Minetti et al., 1999). Spine movement differs between the 49 two types of flight phase. Specifically, while the spine is flexed in collected flight (Fig. 1a), it is ex-50 tended in extended flight (Fig. 1a). This difference of spine movement is crucial for distinguishing 51 between the rotary and transverse gallops. The criterion also provided the mechanism by which the rotary gallop produces higher-speed loco-85 motion than the transverse gallop. We discussed the mechanisms for different spine movements 86 and gait speeds in rotary and transverse gallops based on our findings.

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In this study, we used a two-dimensional physical model consisting of two rigid bodies and two 90 massless bars (Fig. 2). The bodies are connected by a joint, which is modeled to emulate the spine 91 bending movement, and has a torsional spring with a spring constant of . The bars represent the 92 legs. We assumed that the fore and hind parts of the model have the same physical parameters. 93 and are the horizontal and vertical positions, respectively, of the COM of the whole body.  remained unchanged. Therefore, we also neglected the horizontal ground reaction force of the 110 model, which allowed us to ignore the dynamics of , and assumed that the leg bars were always 111 vertical to the ground. We discuss the effects of our assumption about the pitching movement on 112 the galloping dynamics in more depth in Supplementary information S1. 113 During the flight phase, the equations of motion for and are given by (2 + 2 2 sin 2 )̈= −4 − 2̇2 sin 2 .
When the foot touches the ground, it receives the GRF. Because the COM vertical positions are identical between the fore and hind bodies, the foot contact of the fore and hind legs occurs simultaneously. This condition is given by where = [̇̇] ⊤ and * − indicates the state immediately prior to the foot contact. Because the duty factor in animal galloping is small (Hudson et al., 2012), we assumed that the stance phase is sufficiently short and that the foot contact can be regarded as an elastic collision, involving no position change and energy conservation. The relationship between the states immediately prior to and immediately following the foot contact is given bẏ where * + indicates the state immediately following foot contact. The derivation of these equations 114 is presented in Supplementary information S2. 115 In this study, we solved these governing equations under the condition | | ≪ 1 and |̇| ≪ 1. The linearization of the equations of motion (1) gives where = ∕ , = ∕ √ ∕ , = ∕( 2 ), = ∕( ), = ∕ , = √ 2 ∕ , and from now on, * indicates the derivative of variable * with respect to . The foot-contact condition (2) is approximated by where = [̇̇] ⊤ . The foot-contact relationship (3) is linearized by Derivation of periodic solution 116 Rotary galloping involves two flight phases and two stance phases in one gait cycle. In this study, 117 we obtained analytical periodic solutions with two flight phases and two foot contacts for each 118 gait cycle based on the linearized equations (4), (5), and (6) (because transverse galloping has two 119 flight phases and two stance phases in two gait cycles, we assumed this as one gait cycle for the 120 solutions). 121 We defined the periodic solution aŝ( ) = [̂( )̂( )̇̂( )̇̂( )] ⊤ (0 ≤ < 1 + 2 ), where = 0 is the onset time of the first flight phase and 1 and 2 are the durations for the first and second flight phases, respectively. From (4), we obtain where , , > 0, and − ≤ < ( = 1, 2) are constant. We assumed that < 2 ∕ ( = 1, 2) 122 because animals do not oscillate their spines more than once in one gait cycle. To obtain the 123 periodic solution, we have to determine , , , , and ( = 1, 2). 1 and 2 indicate the amplitudes 124 of the first and second spine joint oscillations, respectively. 125 Because the foot-contact condition is satisfied at the first foot contact ( = 1 ) and second foot contact ( = 1 + 2 ), (5) gives From the foot-contact relationship (6) and periodic condition, we obtain From the conditions (8)-(11), we determine ten constants , , , , and ( = 1, 2) to obtain the periodic solution. However, these conditions produced various types of solutions, including solutions that are unlikely in animals. Therefore, we focused on solutions which satisfŷ so that the COM vertical position remained unchanged at each foot contact, as shown in Fig. 3a. 126 Under this assumption, the periodic solution is symmetric with respect to = 1 ∕2 and = 1 + 2 ∕2 127 from the periodic condition, as shown in Fig. 3b. It has been reported that quadruped animals 128 show this symmetric property in locomotion (Raibert, 1986). The symmetry condition (12) forces 129 the third and fourth rows in (11) to be satisfied and reduces two conditions (this mechanism is 130 presented in Supplementary information S3). As a result, the number of independent conditions 131 is reduced to nine. To find a unique solution, another condition (e.g., total energy) is needed.  151 We distinguished types CE and EC with the assumption that the amplitude of oscillation of in the 152 first flight phase is greater than that of the second flight phase ( 1 > 2 ). 153 The periodic solution gives an important criterion to determine the solution type; the signs oḟ− anḋ+ are different for one type of flight (solutions of types C, E, CC, and EE) while they are identical for two different types of flights (solutions of types CE and EC). The criterion gives a hypothesis: while the effect of GRF is too small to change the direction of the spine movement in the gallop with two different types of flights, the effect is so large that the direction changes in the gallop with one type of flight. We evaluated this hypothesis as follows. The difference oḟ+ anḋ − is given by where Δ =̇+ −̇− is the vertical impulse at the foot contact. To investigate the ratio of the angular velocity change to the amplitude of the angular velocity, we define When > 1, solutions have one type of flight. In contrast, solutions with two different flight types 154 have < 1.

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Small gait cycle durations allow animals to kick the ground frequently for acceleration and achieve high-speed locomotion (Hudson et al., 2012). Because short flight durations induce small changes in the COM height, we investigated the COM height changes. From (7), we obtained the COM height changes in the first flight ℎ 1 and second flight ℎ 2 by the difference between the  maximum height at apex and the minimum height at foot contact in each flight as follows: Stability analysis 156 When we found periodic solutions, we computationally investigated the local stability from the 157 eigenvalues of the linearized Poincaré map around the fixed points on a Poincaré section. We 158 defined the Poincaré section by the state just after the second foot contact. Because our model 159 is energy conservative, the gait is asymptotically stable when all the eigenvalues except for one 160 eigenvalue of 1 are inside the unit cycle (these magnitudes are less than 1). 162 To determine the physical parameters ( , , and ) of our model for cheetahs, we used whole-

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Periodic solutions and comparison with measured data 221 To determine the periodic solution in (7), we obtained 1 , 2 , 1 , 2 , 2 , 2 , 1 , and 2 from (8)-(12) as functions of 1 and 1 as follows: 1 ( 1 , 1 ) = 2 ( 1 , 1 ) = 1 − 1 cos 1 , cos 2 ( 1 , 1 ) = 1 cos 1 2 ( 1 , 1 ) , (16e) 1 and 1 satisfy Γ( 1 , 1 ) = 0, where respectively. Figure 6e also shows stable solutions for and 2 for the cheetah model. These stable 246 solutions were compared with the measured animal data. While type EC had a small region for 247 stable solution, the measured data for cheetahs were located close to the stable region, as shown 248 in Fig. 6c and e. In contrast, the measured data for horses were located in the region of stable 249 solution of type E, as shown in Fig. 6d. Furthermore, stable solutions of type EC did not exist in the  The COM height changes in the first flight ℎ 1 and second flight ℎ 2 were obtained as Because ℎ 1 and ℎ 2 monotonically decrease as 1 increases when − ≤ 1 < 0, the COM height 259 change of type EC is smaller than that of type E. When we compared the COM height changes 260 between cheetahs and horses, the periodic solutions showed ℎ 1 = 0.029 and ℎ 2 = 0.10 for the 261 cheetah model and ℎ 1 = ℎ 2 = 0.14 for the horse model. The measured data showed ℎ 1 = 0.008 262 and ℎ 2 = 0.024 for cheetahs and ℎ 1 = ℎ 2 = 0.12 for horses. 263 Changes 1 and 2 of the phase angle of the spine joint angle during the first and second solutions of type EC (Fig. 6c and e). The properties of the measured data of horses were included 281 in those of stable solutions of type E (Fig. 6d). Furthermore, there were not stable solutions of type 282 EC in the range of measured from horses (Fig. 6d).  Fig. 6c and e) and and 1 of the measured data of horses were included in those 293 of stable solutions of type E (Fig. 6d). Our solutions reproduced animal galloping from the view-294 point of spine movement. Furthermore, stable solutions of type EC did not exist in the range of 295 measured from horses (Fig. 6d), as two types of flight never appear in the transverse gallop of 296 horses. These results suggest that the solutions of types EC and E correspond to the rotary and 297 transverse gallop, respectively. Δ̇of the spine movement caused by the GRF and the amplitude 1 of the angular velocity, and 305 obtained the ratio = Δ̇∕ 1 in the obtained solutions and animals. When the direction of the 306 spine movement does not change, < 1 because Δ̇is smaller than 1 . In contrast, when > 1, 307 the direction changes because Δ̇is larger than 1 . We achieved < 1 in cheetahs while > 1 in 308 horses for both the solutions and measured data. Furthermore, Δ̇s were not so different between 309 cheetahs and horses, and 1 of cheetahs was much larger (over six times) than that of horses, as 310 shown in Fig. 8. These results suggest that cheetahs have so large spine movement as to reduce 311 the effect of the GRF, which prevents the direction from changing. This allows cheetahs to create 312 two different flight types. In contrast, horses do not exhibit substantial spine movement, which 313 lets the GRF change the direction. This forces horses to create only one flight type. 315 Small gait cycle durations allow animals to kick the ground frequently for acceleration and achieve 316 high-speed locomotion (Hudson et al., 2012). As shown below (18), COM height changes ℎ 1 and 317 ℎ 2 of cheetahs are both smaller than those of horses, which implies that the flight durations of 318 cheetahs are smaller than those of horses. This result suggests that cheetahs use rotary gallop to 319 create smaller flight durations than those of the horse transverse gallop, which allows cheetahs to 320 move faster than horses. 321 The criterion explained above also provided the mechanism by which the rotary gallop pro-322 duces higher-speed locomotion than the transverse gallop. When cheetahs show both solutions 323 of type EC and type E with identical 1 , solutions of type EC have shorter flight phases than those of 324 type E, as shown in Fig. 7c. This is because while the phase angle change 1 + 2 of the solutions 325 of type EC is almost 2 , that of the solutions of type E is much larger than 2 , which makes the gait 326 cycles of the solutions of type EC shorter than those of the solutions of type E. This result suggests 327 that cheetahs produce high-speed locomotion using rotary gallop rather than transverse gallop. Δ̇s are not so different between cheetahs and horses, and 1 of cheetahs is much larger than that of horses. Therefore, while signs oḟ− anḋ+ are identical for cheetahs, they are different for horses.
This indicates that the second flight is also extended and that solutions of type EC never exist. 341 Finally, we suppose that < − √ .
In this case, we can show that solutions of type CE never 342 exist, in the same way as above. Therefore, when | | > √ , the solutions of type EC and CE never 343 exist due to the foot-contact relationship. involves only collected flight, the measured data were located in the stable solutions of type E but 352 not type C (Fig. 6d). To overcome this limitation, several improvements are needed. For example, 353 our two rigid bodies need to be asymmetric because horses have different physical properties be-354 tween the fore and hind parts of the body, and bend their backs around the sacrum rather than 355 in the middle of the spine (Hildebrand, 1959). In addition, the torsional spring in the spine joint 356 should be asymmetric in the extension and flexion directions because the spine of the horse is dif-357 ficult to bend (Gyambaryan, 1974), particularly in the extending direction (Licka and Peham, 1998). 358 Furthermore, we neglected the dynamics in the horizontal and pitching direction in our model. It  (Taylor, 1978;Alexander, 1988). However, trunk muscles are also effectively used as actua-365 tors to produce energy for acceleration (Hildebrand, 1959;Biancardi and Minetti, 2012). Finally, 366 we also intend to investigate the effect of trunk control on locomotion speed and energy efficiency 367 in the future.

S1 Effect of assumption on galloping dynamics
We ignored the pitching movement of the whole body in our model (Fig. 2) because the COM vertical and spine joint movements are more important for determining galloping dynamics, compared with pitching movements. This assumption induced simultaneous foot contact between the fore and hind legs.
We investigated this dynamical effect based on a model which incorporates θ as the pitch angle of the whole body, as shown in Fig. S1. In this case, the foot contact does not necessarily occur simultaneously between the fore and hind legs.
The motion of this model is governed by the equations of motion of Y , θ, and ϕ, which are given by where immediately following the foot contact is given by where P 1 and P 2 are the impulses of the fore and hind legs, respectively (P 1 > 0 for the foot contact of the fore leg, P 2 > 0 for the foot contact of the hind leg; otherwise P i = 0), and can be determined to satisfy the energy conservation.
We assumed that |θ| ≪ 1, |ϕ| ≪ 1, |θ| ≪ 1, and |φ| ≪ 1. The linearization of the equations of motion (S1) and relationship between the states immediately prior to and immediately following the foot contact When θ = 0, the motions of the fore and hind parts of the model are symmetrical, resulting in simultaneous foot contact between the fore and hind legs and f 1 = f 2 = f (p 1 = p 2 = p). This effect on y and ϕ is identical to that of individual foot contact between fore and hind legs with f 1 = 2f and f 2 = 0 (p 1 = 2p and p 2 = 0) and f 1 = 0 and f 2 = 2f (p 1 = 0 and p 2 = 2p). Therefore, even when we ignore the pitching movement (θ = 0), y and ϕ have no significant effect.

S2 Foot contact dynamics
Here, we derive the relationship (3) between the states immediately prior to and immediately following foot contact in the model. We assumed elastic collision for foot contact that involves no position change and energy conservation. We define ∆ P as the impulse at foot contact from the ground in the vertical direction. ∆ P ϕ is the change in the angular momentum caused by the impulse. The relationship of the translational and angular momentum between immediately prior to and following the foot contact gives From energy conservation, we obtain From (S5) and (S6), we obtain (3).

S4 Parameter dependence of solutions
Here, we show how the types of solutions depend on d and j.
Therefore, only the solutions of types C, CC, and CE, and solutions like Fig. S2b exist when d < − √ j.
This indicates that the second flight is collected. Therefore, solutions of types E and EE never exist.
This indicates that the second flight is extended. Therefore, solutions of types C and CC never exist.
Therefore, only solutions of types CE or EC exist (they are identical because c 1 = c 2 ).

S4.4 When 0 < d < √ j
Only solutions of types E, EE, and EC can exist in the same way as S4.2.

S4.5 When
Only solutions of types E and EE can exist in the same way as S4.1.