P’l//li’12~flllli/l'(ii/llliiPllli’lfd I , 1/1111 ………. • ii ii ii (ZII2,(ll,i “/ill/ill////l/iSl///lilY/////ill Fig. 2. (c) Pace, camel. (a) Walk.
The legs move a quarter period out of turn, in a figure-eight wave.
The amble is the running form (lower duty factor) of this stepping sequence. (b) Trot.
Diagonal legs, ie left front/right back, move together and in phase.
The right front and left back legs move together, half a period out of phase with the other pair.
The trot is a running gait; therefore, the limbs of a trotting animal have duty factors less than 0.5. (c) Pace (or Rack).
The left legs move together and in phase.
The right legs move together, half a period out of phase with the left legs. (d) Canter.
Right front/left back legs move together and in phase.
The left front and right back legs move half a period out of phase with one another and out of phase Coupled Nonlinear Oscillators and the Symmetries of Animal Gaits 355 l// //////////l¢////f | I • • /////~//////////l~//////////~ r[[.,”[.¢/l[[[[/Illlll • • • [ IIII#I/IIIIIIIIIA~IIIlllIIIlll • • …… I • — 373 2 3 4 5 A A A+ ½ ] A+ ½ t [ BIFURCATE A + ½ 1- To~zrri~ A+ ½ ] A I A = ½ period A + ½ [ B = ½ period A A+ ½ A+ ½ a A A+ ½ A+ ¼ I A+¼ [BIFURCATE A l- zo~zrrmF, A+ ½ 1 B + ½ I a = ½ Period B I B = ½ period B B+½ B B B+½ pronk trot similar to rotary gallop to rot y gallop (opposite orientation) bound* pace canter? pronk pace bound* trot pronk bound* walk and amble trot pace asymmetric bound* asymmetric bound* *Bound is close to transverse and rotary gallops. are thinking of physical symmetries of the animal, then this arrangement would be appropriate for quadrupeds whose four legs are approximately the same and where the mechanical coupling between them is relatively similar.
If we are thinking of locomotor central pattern generators then genuine square symmetry is reasonable.
Type 2 has rectangular symmetry, and is appropriate for animals whose fore limbs and hind limbs are fairly similar, but where the left/right coupling differs substantially from the front/rear.
It might be argued that in animal limbs, exact front/back symmetry never occurs.
In this case type 4 is more appropriate.
However, approximate symmetry between front and back is common.
Raibert (1986, 1988) notes that the rotary gallop of the cat is symmetric, apart from minor deviations, under time reversal and such a symmetry in particular interchanges front and back.
Incidentally, front/back symmetry does n o t imply that the animal can move backwards as easily as forwards: this concept refers to the hypothetical interchange of front and back limbs, not to a reversal of direction.
The oscillations of individual limbs can be “‘directional.” 374 J.
Collins and I.
Stewart Type 3 preserves the differences in coupling of type 2 but has more symmetry.
It treats the two front legs as a unit, coupled to an identical unit at the rear.
Its symmetries are independent transpositions of oscillators (12) and (34), together with the interchange of front and rear.
If, for example, the front and rear legs are mainly coupled through the spine, type 3 is appropriate.
In the abstract, type 3 is isomorphic to type 1, which is why their lists of oscillation patterns (Table 3) are very similar.
To be precise, if we renumber oscillators 1234 in type 3 as 1423 (so that front and back pairs in type 3 correspond to diagonal pairs in type 1), then the networks become identical.
We distinguish the two cases because the labeling as limbs is not preserved by this isomorphism, but it means that we can read off the answers for type 3 directly from those for type 1.
In the same way, networks 2 and 5 are isomorphic.
Types 4 and 5 are analogous to 2 and 3, but now the front pair of legs differs substantially from the back pair.
A s Table 3 shows, each type of arrangement has its own particular set of “natural” oscillation patterns.
The most symmetric gaits (pronk, trot, bound, pace, walk) correspond precisely to patterns that occur in the table.
The final gait listed for type 1 has the correct phase relations for a canter, but a true canter does not involve the half-period property of waveform B.
However, minor breaking of the square symmetry could destroy this property, leaving something closer to a canter.
The rotary and transverse gallops are not represented in our list, although type 1 has two conjugate patterns, lc and ld, that are similar to the rotary gallop.
Among the twenty patterns listed for the first three networks (all oscillators identical), only four do not seem to correspond to gaits described in Sect. 2.
All of these involve the half-period condition.
This 2:1 frequency-locking effect and its relevance to animal locomotion are treated in greater detail in Collins and Stewart (1992).
The oscillation patterns for types 4 and 5 are plausible for animals whose front legs are significantly different from their rear legs.
For example, pattern 5b corresponds to a two-legged walk on hind legs, while the front legs move together in phase; pattern 4b is essentially the normal bipedal human walking/running gait with A representing arm movements and B leg movements. 8.
Numerical Simulations In this section we describe some simple numerical simulations that show just how common, and how varied, symmetry-breaking oscillations are, in symmetric networks of identical oscillators.
The model (see equations (3) below) consists of coupled oscillators of van der Pol type, with additional terms that break the “internal” oddfunction symmetry of a conventional van tier Pot oscillator.
We are forced to add such terms because internal symmetries have a strong effect on the entire analysis.
The equations describe four oscillators Oj(j = 1. . . . . 4), each of which involves two dynamic variables (x j, y j).
There are five parameters a , /3, 3′, 8, e, of which oz, /3, and e affect the internal dynamics of each oscillator, and 3, 3′ are coupling constants.
We have chosen a simple form of linear coupling.
The model has no special physiological significance since it is presented only as evidence that the mathematical phenomena that we have described are easily observed in actual equations.
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