## Problems Of Diagonal Sequence Walks

Most primates preferentially employ the DS footfall sequence in walking (Hildebrand, 1967, 1985; Prost, 1965, 1969, 1970; Rollinson and Martin, 1981; Vilensky, 1989; Vilensky and Larson, 1989). The primate preference for DS walks is problematic, because (as many people have shown) such gaits

(B) Duty factor

Figure 2. (Continued) (B) variation in the y-axis only (walking gaits, duty factor = 74). The Y variable, here called diagonality, represents the percentage of time in the gait cycle by which each fore footfall lags behind the fall of the hindfoot on the same side. The four illustrated gaits differ in the phase relationship between the forelimbs (shown in the same position in all four drawings) and the hindlimbs. When both feet on one side are in synchrony in walking (diagonality = 0 or 100), the gait is a walking pace. When they are exactly out of phase (diagonality = 50), the gait is a walking trot. When the fore footfall lags the ipsilateral hind footfall by more than zero but less than 50% of the cycle period, the gait is a lateral-sequence walk; when it lags by more than 50% but less than 100%, the gait is a DS walk. The right hindlimb is shaded in the four drawings to emphasize the phase shift.

(B) Duty factor

Figure 2. (Continued) (B) variation in the y-axis only (walking gaits, duty factor = 74). The Y variable, here called diagonality, represents the percentage of time in the gait cycle by which each fore footfall lags behind the fall of the hindfoot on the same side. The four illustrated gaits differ in the phase relationship between the forelimbs (shown in the same position in all four drawings) and the hindlimbs. When both feet on one side are in synchrony in walking (diagonality = 0 or 100), the gait is a walking pace. When they are exactly out of phase (diagonality = 50), the gait is a walking trot. When the fore footfall lags the ipsilateral hind footfall by more than zero but less than 50% of the cycle period, the gait is a lateral-sequence walk; when it lags by more than 50% but less than 100%, the gait is a DS walk. The right hindlimb is shaded in the four drawings to emphasize the phase shift.

appear to be inherently less stable at slow speeds than typical LS walks. In the LS walk of a horse or other typical mammalian quadruped, the areas of the tripods of support (gray triangles, Figures 3A, 4-5, 8-9) are maximized, with three support points well spread out along the anteroposterior axis (cf. Figure 1). The vertical line through the animal's center of mass (line of gravity)

probably falls inside the triangles during most of the tripedal support phases. In the bipedal phases of the cycle, when the line of gravity necessarily gets off the line of support, the animal tends to roll to the left (Figures 3A) or right (Figures 3A, 7), but the next foot to come down descends in the right place to check the roll (Figure 3A, nos. 4, 8).

The DS walks seen in primates are considerably less stable. Because the hindfoot in such walks strikes down close behind the forefoot on the same side, the unilateral bipod of support (support by only two feet on the same side) is extremely short, and so the tripods or triangles of support are much smaller than they are in an LS walk (Figure 1; Figure 3A, nos. 4, 8, Figure 3B, nos. 6, 8). Depending on the values of certain gait parameters, the animal may have to balance briefly on the small unilateral bipod twice in each gait cycle (Figure 3B, no. 7; cf. Figure 4). The periods of instability in a DS walk do not appear to contribute uniformly to forward movement. In fact, the direction of pitch may be slightly toward the rear when the forefoot comes down (Hildebrand, 1980), so that the animal is periodically on the verge of toppling backward.

It is not clear what advantage DS walks confer that compensate for these apparent disadvantages. Few answers have been suggested. None of them are persuasive, and none of them satisfactorily account for the observed distribution of LS and DS gaits among mammals.

Muybridge (1887) proposed: (1) that the stronger limb on each side always descends immediately before the other (his "Law of the Walk"), and (2) that arboreal climbing has given primates exceptionally strong fore-limbs—and hence DS gaits. But as Vilensky and Larson (1989) observe, there is no reason to think that primate forelimbs are stronger, or bear greater stresses, than their hindlimbs. Force-plate studies suggest precisely the reverse (Demes et al., 1994; Kimura, 1985, 1992; Kimura et al., 1979; Lemelin and Schmitt, this volume; Reynolds, 1985; Schmitt and Lemelin, 2002).

Prost (1965) suggested that DS gaits, but not LS gaits, allow a mammal "to use lateral spine bending to increase distance between successive contact points for the same leg" and thereby to increase stride length. At least some primates do in fact increase stride length in this way (Dykyj, 1984; Demes et al., 1990; Shapiro et al., 2001). But so do some nonprimate tetrapods that use LS walking gaits (Carlson et al., 1979; Pridmore, 1992; Bitter, 1995). What a quadruped needs to allow lateral vertebral flexion to enhance stride

(A) Lateral sequence

(A) Lateral sequence

1-3. Diagonal bipod: Roll left, pitch forward

4-5. LF-RF-LH tripod: Stable

6-7. Unilateral bipod: Roll right

8-9. LF-RH-LH tripod: Stable

10-12. Diagonal bipod (Repeat sequence w/ sides reversed)

1-3. Diagonal bipod: Roll left, pitch forward

4-5. LF-RF-LH tripod: Stable

6-7. Unilateral bipod: Roll right

8-9. LF-RH-LH tripod: Stable

10-12. Diagonal bipod (Repeat sequence w/ sides reversed)

1-5. Diagonal bipod: Roll left, pitch forward

6. LH-RH-RF tripod: Roll left, pitch forward

Figure 3. Support polygons in typical LS (A) and DS (B) walking gaits. The diagrams represent foot placement positions during the first half of a gait cycle, beginning with the onset of the LH-RF diagonal bipedal support phase ("bipod"), as viewed from above. Triangles of support are shown in gray. The second half of the cycle, beginning with position 10 in (A) and 9 in (B), would be the mirror image of the first half. In both sequences, time intervals between successive diagrams are roughly constant. The position indicated for the vertical projection through the center of mass (M) is an approximation based on three assumptions: (1) the animal is roughly in balance at the beginning of the diagonal bipod, (2) the center of mass remains in the midline, and (3) the center of mass moves forward at a constant velocity. Footfall timing and spacing are based on Muybridge (1887, p. 28, Pl. 143); graphic convention after Rollinson and Martin (1981) Symbols: black figures, left foot placement (LH, left hind; LF, left fore); white figures, right foot placement (RF, right fore; RH, right hind); squares, hindfeet; circles, forefeet.

1-5. Diagonal bipod: Roll left, pitch forward

6. LH-RH-RF tripod: Roll left, pitch forward

Figure 3. Support polygons in typical LS (A) and DS (B) walking gaits. The diagrams represent foot placement positions during the first half of a gait cycle, beginning with the onset of the LH-RF diagonal bipedal support phase ("bipod"), as viewed from above. Triangles of support are shown in gray. The second half of the cycle, beginning with position 10 in (A) and 9 in (B), would be the mirror image of the first half. In both sequences, time intervals between successive diagrams are roughly constant. The position indicated for the vertical projection through the center of mass (M) is an approximation based on three assumptions: (1) the animal is roughly in balance at the beginning of the diagonal bipod, (2) the center of mass remains in the midline, and (3) the center of mass moves forward at a constant velocity. Footfall timing and spacing are based on Muybridge (1887, p. 28, Pl. 143); graphic convention after Rollinson and Martin (1981) Symbols: black figures, left foot placement (LH, left hind; LF, left fore); white figures, right foot placement (RF, right fore; RH, right hind); squares, hindfeet; circles, forefeet.

Figure 4. Possible adaptive value of diagonal-sequence walking gaits in primates. At the moment of forefoot touchdown, when weight is about to be transferred to a new and untested substrate, the line of gravity (gray arrow: the vertical through the body's center of mass, estimated here as the vertical through the midpoint of an ischium-to-occiput line) will lie much closer to the supporting hindfoot (gray tone) in the D-S walk of the baboon (A) than the L-S walk of the horse (B). In primates or other arboreal animals with marked grasping specializations of the hindfoot, the primate support pattern allows the animal to draw back or regain its balance if the new support breaks or bends precipitously. (Drawings after Muybridge, 1887; from Cartmill et al., 2002)

Figure 4. Possible adaptive value of diagonal-sequence walking gaits in primates. At the moment of forefoot touchdown, when weight is about to be transferred to a new and untested substrate, the line of gravity (gray arrow: the vertical through the body's center of mass, estimated here as the vertical through the midpoint of an ischium-to-occiput line) will lie much closer to the supporting hindfoot (gray tone) in the D-S walk of the baboon (A) than the L-S walk of the horse (B). In primates or other arboreal animals with marked grasping specializations of the hindfoot, the primate support pattern allows the animal to draw back or regain its balance if the new support breaks or bends precipitously. (Drawings after Muybridge, 1887; from Cartmill et al., 2002)

length is not a DS gait, but a diagonal-couplets gait, in which diagonally opposite limbs swing forward and back as a pair. A diagonal-couplets gait (25< diagonality <75) can be either DS (diagonality >50) or LS (diagonality <50; Figure 2B). Moreover, as Vilensky and Larson (1989) point out, if Prost's (1965) analyses were correct, it is hard to see why other quadrupeds would not adopt DS gaits in order to enhance stride length in walking. This objection can be put more globally: any theory that proposes a benefit accruing to

DS walking gaits needs to explain why most nonprimates have not availed themselves of this benefit.

Prost (1969) subsequently observed that the apparent inferiority of DS gaits (Figure 3) is irrelevant in arboreal locomotion on a narrow branch, since no triangles of support can be formed if all footfalls are collinear. This is an important observation, which refutes some of the supposed adaptive barriers to the adoption of DS walking gaits, but it does not suggest any positive advantage to adopting them. Prost proposed that DS walks are more advantageous than LS walks for an arboreal animal walking on a thin horizontal branch because they allow diagonally opposite limbs to act in concert, and thus reduce rolling and yawing forces during locomotion. Unfortunately, this analysis again confuses diagonal sequences (RF footfall follows LH) with diagonal-couplets (RF and LH move more or less together). It also fails to explain why many arboreal nonprimates use LS gaits.

Most subsequent analyses of the significance of DS walking gaits have argued that primates have DS gaits because they carry a larger percentage of their weight on the hindlimbs than other mammals do. This idea originated with the work of Tomita (1967). Tomita reasoned that a walking animal swinging the RF and LH limbs forward as a pair (diagonal-couplets) might be expected to put them down in a sequence that depends on the position of the animal's line of gravity: forefoot first (LS walk) to stop forward pitch if the line of gravity passes in front of the line connecting the other two, supporting feet (LF + RH, in this case), and hindfoot first (DS walk) to stop backward pitch if the line of gravity passes behind that line. By heavily loading the hindquarters of dogs, Tomita was able to induce DS walking gaits in a small percentage of trials. He concluded that primates use DS walks because they are tail-heavy compared to nonprimates—which accordingly use LS walks instead.

We believe that Tomita's theory contains a fundamental mistake. In an LS walk, the fore footfall at the end of a diagonal bipedal support period does indeed descend in a position where it effectively checks forward pitch (Figure 3A, nos. 3-4). But the corresponding hind footfall in a typical DS walk descends immediately behind, or alongside of, the forefoot in the diagonal support pair (Figures 3B, nos. 5-6)—and so it is not in a position to arrest backward pitch effectively. Moreover, the direction of pitch in a DS walk at the moment of hindfoot touchdown appears to be forward, not backward (Figures 3B, no. 6; Hildebrand, 1980).

Figure 5. Two ways to maximize hindlimb protraction at the moment of forefoot touchdown. (A) the diagonal-sequence, diagonal-couplets pattern, with a diagonality slightly exceeding 50; (B) the lateral-sequence, lateral-couplets pattern, with a diagonality slightly exceeding zero. Baboon A is traced from a photograph by Muybridge (1887, Pl. 143). Baboon B is an artificial construct that has its diagonality lowered by 50—that is, through a 180° phase shift.

Figure 5. Two ways to maximize hindlimb protraction at the moment of forefoot touchdown. (A) the diagonal-sequence, diagonal-couplets pattern, with a diagonality slightly exceeding 50; (B) the lateral-sequence, lateral-couplets pattern, with a diagonality slightly exceeding zero. Baboon A is traced from a photograph by Muybridge (1887, Pl. 143). Baboon B is an artificial construct that has its diagonality lowered by 50—that is, through a 180° phase shift.

Although the symmetry required by Tomita's analysis does not exist, his insights have influenced subsequent thinking about primate locomotion. From his computer simulations of quadrupedal gaits, Yamazaki (1976) reportedly concluded that DS gaits reduce roll in walking—if and only if the hindlimbs bear most of the body weight (Kimura et al., 1979). Unfortunately, Yamazaki's dissertation research has never appeared in print. Kimura and his coworkers, who used force-plates to measure reaction forces on the fore- and hindlimbs of primates and dogs (Kimura et al., 1979), found that vertical forces in primates were greater on the hindlimb than on the forelimb, but that the reverse was the case in dogs. Similar differences have been found in other force-plate studies (Demes et al., 1994; Kimura, 1985, 1992; Reynolds,

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