Many-legged maneuverability: dynamics of turning in hexapods

Sprawled-postured arthropods can be both highly statically stable and remarkably maneuverable. The common perception of a reciprocal relationship between stability and maneuverability is false for animals such as rapidly running cockroaches and crabs. Although many-legged, sprawled-postured arthropods can be highly statically stable (i.e. the center of mass lies within at least a tripod of support; Alexander, 1971), this static stability does not preclude dynamics from influencing locomotion (Ting et al., 1994). Force, velocity and inertia are important for maintaining stable motion, particularly at high speeds. For example, ignoring the dynamics of many-legged, sprawled-postured arthropods would have prevented the discovery that diverse legged animals can all be modeled by a simple spring-mass system (Blickhan and Full, 1987; Full, 1989; Full and Tu, 1990, 1991). Bipedal runners and hoppers as well as four-, six- and eight-legged trotters maintain a stable, re-entrant path of their center of mass in the vertical plane of forward motion from step to step in a very similar way (Alexander, 1990; Blickhan, 1989; Blickhan and Full, 1993; Cavagna et al., 1977; Farley et al., 1993; Full and Farley, 1999; McMahon and Cheng, 1990). Diverse morphologies can all employ bouncing dynamics during straight-ahead running.

The generality of spring-mass dynamics naturally raises the question of when leg morphology influences locomotion. The advantages and disadvantages of increased leg number and sprawled posture during running remain speculative. However, we do know that spring-mass dynamics is not attained by using all the legs in a similar manner. During constant average-speed, straight-ahead running, cockroaches operate like a spring-mass system in which three legs sum to operate as one leg of a biped or two legs of a quadruped. Each leg of the tripod (a front leg and hind leg on one side and a middle leg on the other) functions differently (Full et al., 1991). The front (prothoracic) pair of legs only decelerates the insect during the stance phase, while at the same time the hind (metathoracic) pair of legs only accelerates the animal forward. The middle (mesothoracic) pair of legs works much like human legs. The middle pair first decelerates and then accelerates the body during a step. Ground-reaction forces tend to align along the axis of each leg, minimizing joint torque (Full et al., 1991).

We propose that differential leg function and morphology also enable statically stable, many-legged animals to be remarkably maneuverable. The present study examines turning, a maneuver that takes place primarily in the horizontal plane. Two principal turning strategies have been proposed for insects: increasing step frequency (Graham, 1972; Zolotov et al., 1975) or step length (Franklin et al., 1981; Frantsevich and Mokrushov, 1980; Graham, 1972; Jander, 1985; Strauß and Heisenberg, 1990; Zollikofer, 1994; Zolotov et al., 1975) of legs on one side of the body relative to the other. At the extreme of changes in step length, some insects may fix one leg in place (Frantsevich and Mokrushov, 1980; Zolotov et al., 1975) and pivot about it, or move the legs on one side of the body backwards (Franklin et al., 1981; Frantsevich and Mokrushov, 1980; Zolotov et al., 1975). Additional changes in step geometry, such as alterations in the functional length of the legs, have also been observed (Franklin et al., 1981).

Changes in stride frequency or stride length must act to turn the body by causing translational and rotational accelerations that result in the alterations in heading (the direction of the velocity vector of the center of mass) and body orientation observed during a turn. Legs may act in different ways to produce these accelerations during turning (Hughes, 1989), analogous to the differential leg function observed during straight-ahead running. Camhi and Levy (1988) addressed the possibility that motions of some legs during turning may reflect active contributions to turning and that motions of other legs may be due to motion of the body relative to the position of the foot on the substratum. This possibility underscores the fact that kinematic measurements alone may not reflect the different contributions of individual legs to moving the body, as has been emphasized in several studies of maneuverability in arthropods (Camhi and Levy, 1988; Cruse and Silva Saavedra, 1996; Domenici et al., 1998; Hughes, 1989; Land, 1972). Recent research (Domenici et al., 1998) has made progress towards separating leg movements due to active contributions by legs, or sets of legs, to turning from movements due to body motion in unrestrained, intact animals. We contend that it is necessary to add a consideration of kinetics (force and inertia) to kinematic measurements to understand the potential benefits of the turning strategies used by many-legged animals. Understanding the full movement dynamics also allows us to generate hypotheses about how leg and body morphology and function affect potential turning performance.

We take the next step in elucidating the mechanisms of maneuverability by measuring the full dynamics of turning. Executing a successful turn involves at least two requirements. First, the animal’s mean heading must change from an initial direction during straight-ahead running to a new direction (Fig. 1A). This angle of deflection of the center of mass (COM) trajectory (θ_d) depends on the magnitude of the perpendicular force impulse (F_p, the force perpendicular to the initial heading, integrated over the step period) relative to the forward momentum of the COM (the product of body mass and forward velocity).

A second requirement for a successful turn involves rotating the fore–aft axis (represented by a line through the COM and the head) so that it aligns with the heading (Fig. 1B). To minimize the degree of misalignment between deflection and body rotation, the torque impulse (net torque integrated over the step period) relative to the inertia of the body about the yaw axis must cause a rotation (θ_r) equal to the heading deflection. The specific objective of the present study was to determine the forces and torques necessary to produce a turn and the legs responsible for force and torque production. We chose to study turning in the death-head cockroach Blaberus discoidalis because of the complete, three-dimensional, dynamic data already available for straight-ahead running (Blickhan and Full, 1987; Full and Ahn, 1995; Full et al., 1991, 1998; Full and Tu, 1990, 1991; Kram et al., 1997). From the two-dimensional kinematics of turning, we estimated the net perpendicular forces and rotational torques on the COM. Since many-legged animals, such as cockroaches, typically have more than one leg on the substratum at one time, the force and torque contributions of individual legs to the net force and torque on the COM are indeterminate. By estimating single-leg ground-reaction forces using a photoelastic technique (Full et al., 1995), we determined which legs were responsible for the force and torque production necessary to turn the body. We interpreted the data using a simple, general model which relates leg function (i.e. force production) and morphology (i.e. leg position) to turning performance.

We used the death-head cockroach Blaberus discoidalis (Serville), with a mass of 4.6±0.26 g (mean ± S.D., N=6). Cockroaches were housed in plastic containers and fed dog food and water ad libitum.

We used a photoelastic technique to measure single-leg forces from freely moving cockroaches (Full et al., 1995). The light source for all trials was a projector (Dukane, model 653, with a 360 W bulb). We constructed two large gelatin plates by gluing pieces of glass 0.32 cm thick onto a 40.5 cm×40.5 cm square glass bottom to form a square well G0.32 cm deep. One plate had a 28.6 cm×28.6 cm square well, and the second plate had a 24.5 cm×24.5 cm square well. A linear polarizer (a sheet of Polaroid HN38S) was attached to the bottom of the plates so that the polarizing axis was parallel to one side of the well. A lens-mounted polarizer (Edmund Scientific photo mount polarizing filter) served as an upper filter. We constructed an acrylic track with a 15.5 cm deep × 10 cm wide × 71 cm long runway leading to a 15.5 cm deep × 21 cm wide × 25.4 cm long open area. The gelatin plates formed the floor of the open area and were held by the track 22.5 cm above the ground and 2.54 cm above the projector. Fiber-optic lights provided epiluminescence. Epiluminescence did not affect the appearance of the gelatin or the stress-induced light patterns.

The track was constructed so that the cockroaches emerged from a relatively narrow chute into the open area and onto the gelatin surface, where they were confronted by a curved or slanted cardboard wall in their direction of motion. The cockroaches had to turn or stop to avoid hitting the wall. Towards the rear of the open area was a darkened refuge where the cockroaches could retreat after turning.

We prepared the gelatin slabs in the same manner as described by Full et al. (1995). During video-taping, we moistened and cleaned the gelatin by wiping it with lint-free tissue and a mild soap solution. After wiping the gelatin, we allowed it to dry until the gelatin was moist enough that the cockroaches neither slipped on the gelatin nor stuck to it.

We video-taped trials from above at 500 frames s⁻¹ using a high-speed camera (Kodak Ektapro 1000 with an image intensifier or a Redlake Motionscope). The camera was positioned above the track to make its field of view approximately 12 cm×10 cm. A field of view that is as small as possible maximizes the resolution of the video for the most accurate gelatin force recordings. However, the field of view must also be large enough to allow multiple steps within the field of view. Video-taped segments were then transferred to computer files using a frame-grabbing board (Neotech Image Grabber, NUBus) and software (program implemented using Ultimage Concept V.i., Graftek, France, package for LabVIEW 2.2.1, National Instruments).

Cockroaches were not trained prior to testing. The animals were stimulated to run down the track by lightly touching their cerci. All trials were video-taped, but only trials in which the cockroach turned in the field of view of the camera, without slipping or sticking to the gelatin, were considered for analysis. Of these, we only selected trials for analysis in which the legs of the animal and the optical patterns were minimally obscured by the body. To qualify as a turn produced during running, the fore–aft axis angle had to change by more than 20 ° and the initial velocity had to be greater than 15 cm s⁻¹.

The probability of a successful trial was approximately 1 %. Many factors prevented analysis of trials. Cockroaches preferred to stay close to the walls while running (Camhi and Johnson, 1999) and often decelerated and stopped. Their legs and body frequently hit the wall. Moreover, the walls distorted or obscured the optical patterns produced by their tarsi.

We accepted eleven trials from six animals. From these trials, we analyzed the force developed by 41 steps (i.e. tripods), consisting of 114 individual leg steps. In total, 3560 individual leg force patterns from 1584 images were digitized. Seven trials consisted of at least two strides (four steps) and were used for kinematic analysis.

We used a motion-analysis system (Peak Performance Technology, Peak 5 and Motus) to determine the two-dimensional kinematics of turning. We digitized the head, the midpoint on the axis between the head and the end of the abdomen or ‘tail’ and tail points over the course of the trial. These data were filtered using a fourth-order Butterworth low-pass filter with zero phase shift (LabVIEW 4.0.2 program, National Instruments) at a cut-off frequency of 25 Hz. The fore–aft axis angle (the angle between the fore–aft axis and the global x-axis, defined to be the constant horizontal axis of the video image; Fig. 2A) was calculated from the head and tail points. If the head or tail point of the animal moved outside the field of view, the fore–aft axis angle was calculated from the midpoint and the head or tail point in the field of view. We estimated the position of the COM as a point 54 % of the body length along the fore–aft axis from the head (Kram et al., 1997).

We digitized tarsal or ‘foot’ touchdown positions by approximating the position of the center of the optical pattern produced by the legs on the gelatin during stance. Since the feet did not slip on the gelatin, their position remained constant during a step. We used the average of several measurements of touchdown position over the step as a constant value for foot touchdown position for subsequent analysis.

Linear velocity, linear acceleration, rotational velocity and rotational acceleration were calculated from the positions of the COM and the fore–aft axis angle. We used a fourth-order difference equation (Biewener and Full, 1992) to differentiate position measurements to yield velocity and acceleration. Curvature was calculated using three successive positions of the COM.

We selected a period of each trial (the turn duration) that contained a change in fore–aft axis angle greater than 20 ° and an integral number of steps. We did not select trials on the basis of heading deflection. The magnitude of fore–aft axis rotation (θ_r) was calculated as the difference between the fore–aft axis angle at the end of the period and the fore–aft axis angle at the beginning of the period.

We calculated phase relationships between legs during the steps of a turn by calculating the time at which a leg contacted the surface relative to the step of a reference leg normalized to the period of a reference leg (Jamon and Clarac, 1995). Phase values of 0 or 1 represent simultaneous (in-phase) stepping, and phase values of 0.5 represent alternating (anti-phase) stepping.

We measured foot placements at the beginning of each step (anterior extreme position; AEP) and at the end of each step (posterior extreme position; PEP) in polar coordinates (r, ϕ). The origin was placed at the COM, and ϕ was measured relative to the fore–aft axis (Fig. 2B). For comparison, we also calculated AEPs and PEPs for straight-ahead running from five three-dimensional kinematic data sets previously collected in our laboratory (Kram et al., 1997). The AEP was estimated as (r, ϕ) of the tibia–tarsus joint when the angle ϕ was minimal, and the PEP as (r, ϕ) when ϕ was maximal. We assumed that foot placement during straight-ahead running is symmetrical about the midline, and pooled the data from the right and left sides of the body, resulting in 10 measurements for each leg. Since the animals used in the present study were approximately 30 % longer than the animals in the previous study, we scaled the r components of the AEP (r_AEP) and the r components of the PEP (r_PEP) from the previous study by 1.3 when comparing them with the positions calculated during turning.

The optical patterns during complete steps deviated from the optical patterns modeled in Full et al. (1995). Discrete lobes of the ‘cloverleaf’ were often difficult to see at the beginning and end of the step (when the forces were smaller than during mid-stride), and patterns deviated from the expected elliptical shape even when force production was maximal for the step. In analyzing all patterns, we attempted to estimate the center of the pattern (which was made easier by considering the preceding and subsequent patterns and assuming that the point of contact was the same) and collected area and intensity values from the four quadrants surrounding the estimated center of the pattern (the origin).

When portions of the optical pattern became obscured by the body of the cockroach, we filled in the obscured area by eye or by reflecting the unobscured half of the lobe’s midline to the obscured half. In cases in which the entire pattern passed under the body, we did not collect forces from the pattern during that time. These periods usually occurred at the end of a step, when forces were relatively small, and did not affect the conclusions of our analysis.

Large horizontal forces also make it difficult to measure vertical forces because, in situations where horizontal forces are large enough to cause the quadrant ratio (the ratio of the larger half of the pattern to the smaller half; see Fig. 5 in Full et al., 1995) to reach 1, the vertical force becomes indeterminate and cannot be measured. Vertical forces obtained by analyzing patterns where the quadrant ratio was 1 represent estimates of minimum vertical forces. We did not use vertical force measurements in the present analysis, but this limitation should be taken into consideration in future applications of the technique.

Resultant horizontal forces calculated by the LabVIEW program were resolved into global components based on an axis of the video field. The global x-axis (Fig. 2A) was set to coincide with the horizontal axis of the video field (see Fig. 5 and equation 7 in Full et al., 1995). These components were filtered using a cut-off frequency of 100 Hz. The magnitude of the filtered resultant horizontal force and the force angle were then recalculated from the filtered components.

We determined the contribution of individual legs to deflection of the heading by resolving the resultant horizontal force into components relative to the heading of the animal (Fig. 2D). The component parallel to the heading of the body (F_h) accelerates or decelerates the body. The component perpendicular to the heading (F_p) changes the heading direction. We calculated the perpendicular force impulse (ΔP_leg) by integrating F_p with respect to time over the duration of the step period.

We determined the contributions of individual legs to rotation of the body by calculating torque (T) about the COM produced by each leg (Fig. 2D). T was calculated as the component of the horizontal force perpendicular to the moment arm (the vector from the COM to the line of action of the force) times the length of the moment arm (l). Positive T acts to rotate the body in a counter-clockwise direction. We calculated the torque impulse generated by a single leg (ΔL_leg) by integrating T with respect to time.

We compared ΔP_leg and ΔL_leg among individual legs over periods limited to an integral number of strides. To control for differences in turn magnitudes, which may change the magnitudes of the total forces produced by the legs, we ranked ΔP_leg and ΔL_leg among the legs in a given tripod.

We calculated the mean direction of ground-reaction forces relative to the heading by resolving the resultant horizontal forces into their components F_h and F_p, averaging the F_h and F_p measurements over the course of the step, then calculating the mean force direction as tan⁻¹[(mean F_p)/(mean F_h)].

We used Mann–Whitney U-tests to compare variables calculated from kinematic measurements (i.e. θ_d, the force impulse in the direction perpendicular to the initial heading, ΔP, the torque impulse, ΔL, the net force and T). We compared phase relationships between leg pairs with the expected value for antiphase stepping (0.5) using unpaired t-tests. To compare duty factor values among legs, we conducted a nested analysis of variance (ANOVA) with leg and individual as treatments and with leg nested within individual. We used multivariate analysis of variance (MANOVA) to compare leg AEP and PEP values. r_AEP, r_PEP and the corresponding angles between the fore–aft axis and the AEP (ϕ_AEP) and PEP (ϕ_PEP) were all included as dependent variables in the analysis. To compare turning with straight-ahead running, leg and behavior (turning or straight-ahead running) were included as treatments, and the leg/treatment interaction was tested for significance. To compare contralateral sides of the body during turning, leg/side interactions were tested for significance. We used Cochran’s Q-test (Siegel, 1956) to analyze single-leg force production frequency data and Kruskal–Wallis tests to compare leg torque impulses. We used a statistical program (StatView 4.5, Abacus concepts, Inc.) to calculate t-tests, Mann–Whitney U-tests and Kruskal–Wallis tests. We used a separate statistical program (SuperANOVA, Abacus concepts, Inc.) to conduct ANOVA and MANOVA calculations. We calculated Cochran Q-tests manually using a spreadsheet (Microsoft Excel).

Cockroaches employed an alternating tripod gait throughout the course of a turn (Fig. 4A). Phase relationships among contralateral and ipsilateral legs were not significantly different from 0.5 (t-tests, P>0.15 in all cases; Table 1). Duty factors averaged 0.56±0.1 (mean ± S.E.M., N=10) and did not differ significantly between legs (MANOVA, P=0.16).

We selected trials in which the animals ran at close to their preferred speed of 20 cm s⁻¹ (Full and Tu, 1990; Table 2). A trial averaged 3.6 steps. The mean duration of a single step was 69 ms. Cockroaches did not follow a constant trajectory while turning, but accelerated and decelerated during a turn (Fig. 4B,C). Heading change (θ_d) during a complete trial averaged 48 ° (Table 2). The maximum θ_d per step in the trial turn direction was not significantly different from the mean net deflection found for a complete trial (Mann–Whitney U, P=0.92; Table 2). This apparent discrepancy can be explained by the observation that, for many steps, θ_d was directed against the net deflection for the trial. Deflection in one direction tended to alternate with deflection in the opposite direction. Curvature (1/R) varied considerably within trials (Fig. 4D). Periods of relatively straight running were interspersed with periods in which the center of mass was rapidly accelerated perpendicularly, resulting in increased curvature values.

Fore–aft axis change (θ_r) ranged from a minimum of 22 ° in four steps over 230 ms to a maximum of 73 ° in five steps over 412 ms. θ_r per step ranged from 5.5 to 20.4 °. The fore–aft axis remained close to the heading (Fig. 4E), on average lagging behind the heading by 5 ° (Table 2). Rotational velocity varied within trials (Fig. 4F; Table 2). However, the direction of fore–aft axis rotational velocity did not change frequently during turns. The rotational velocity seldom decreased below zero.

The angle between foot placement at the AEP and the fore–aft axis (ϕ_AEP; Fig. 2B) did not differ significantly between straight-ahead running and during turning (MANOVA, P=0.07; Fig. 5). ϕ_AEP values for contralateral legs were not significantly different from each other during turning (MANOVA, P=0.40).

The distance from the center of mass to the foot at the AEP (r_AEP; Fig. 2B) for the inside hind leg was significantly shorter during turning than during straight-ahead running (1.38 cm versus 1.66 cm; MANOVA, P=0.046; Fig. 5). r_AEP for the outside front leg was significantly greater during turning (3.29 cm versus 2.97 cm, respectively; P=0.025), but r_AEP during turning was not significantly different from that for straight-ahead running for any of the other legs. r_AEP values for contralateral legs were not significantly different during turning (MANOVA, P=0.18).

The angle between foot placement at the PEP and the fore–aft axis (ϕ_PEP) for the outside middle leg was significantly greater during turning than during straight-ahead running (95 ° versus 82 °; MANOVA, P=0.027). ϕ_PEP values were not significantly different between turning and straight-ahead running for any other leg. However, ϕ_PEP values were significantly greater for all outside legs relative to the contralateral inside legs during turning (MANOVA, P<0.01 in all cases).

The distance from the center of mass to the foot at the PEP (r_PEP) was significantly smaller for the inside front (2.54 cm versus 2.06 cm) and middle (1.91 cm versus 1.42 cm) legs during turning and significantly greater for the outside hind leg when compared with straight-ahead running (2.59 cm versus 2.16 cm; MANOVA, P<0.05 in all cases). r_PEP values for the outside middle and hind legs were significantly greater than for the contralateral inside legs (MANOVA, P<0.001 in both cases), but contralateral front leg r_PEP values were not significantly different (MANOVA, P=0.22).

Perpendicular impulses (ΔP) which deflect the heading and reflect trajectories of increased curvature (Fig. 4D) varied over the course of a turn. A ΔP which could deflect the body by as much as the total ΔP observed for the entire trial could be generated in a single step (Table 2). The mean ΔP for the step that generated the maximum ΔP among the steps of the trial was not significantly different from the mean value per step or per trial (Mann–Whitney U, P>0.65 in both cases; Table 2). Greater deflection of the heading over the complete trial did not occur because steps producing 70 % of the ΔP towards the turn were observed to be interspersed with steps that produced ΔP against the turn (Table 2). The maximum net perpendicular force (the sum of the F_p produced by all the legs over the duration of the step) exerted on the COM per step was significantly greater than values for the complete turn (Mann–Whitney U, P<0.001). The net perpendicular force per step on the center of mass in the direction of the turn was also significantly greater than values for the complete turn (Mann–Whitney U, P=0.005; Table 2). The mean, maximum net perpendicular force during a step was 3.7 times greater than the mean value per trial (Table 2).

The tripod consisting of the inside front, outside middle and inside hind legs generated ΔP in the trial turn direction in 35 % of the steps made by this tripod (Table 3). The tripod consisting of the outside front, inside middle and outside hind legs generated ΔP in the trial turn direction in 85 % of the steps made by this tripod. The mean magnitude of ΔP in the trial turn direction produced by this tripod was more than twice that of the alternative tripod, resulting in a significant difference in magnitude (MANOVA, P=0.002).

Torque impulses (ΔL) resulting in rotational acceleration which can alter the body angle varied over the course of a turn (Fig. 4G). Maximum ΔL for a single step during a trial was significantly greater than values for the complete trial (Mann–Whitney U, P=0.03; Table 2). ΔL per step was of similar magnitude towards and against the trial turn direction, resulting in small torque impulses per trial (Table 2). The mean net torque exerted on the body per step for steps with net torque in the trial turn direction was significantly greater than values for the complete trial (Mann–Whitney U, P=0.001; Table 2). As for torque impulses, net torques per step were of comparable magnitude towards and against the trial turn direction. Therefore, small net torques were produced during the complete turn (Table 2).

The tripod consisting of the inside front, outside middle and inside hind legs was associated with ΔL in the trial turn direction in 40 % of the steps made by this tripod (Table 3). The tripod consisting of the outside front, inside middle and outside hind legs was associated with ΔL in the trial turn direction in 57 % of the steps made by this tripod. Torque impulse magnitudes did not differ significantly during the steps of alternative tripods (MANOVA, P=0.67).

Individual leg forces perpendicular to the heading were directed most often towards the body (Table 4; Figs 3C, 6). Inside legs produced forces with negative angles relative to the heading direction and trial turn direction (Table 4), indicating that the inside legs directed their forces against the trial turn direction. Outside legs produced forces with positive angles, indicating that the outside legs directed their forces towards the trial turn direction.

Legs on the outside of the turn most frequently generated ΔP_leg which acted to deflect the heading in the trial turn direction compared with legs on the inside of the turn in the same tripod (Cochran Q-test, P<0.001 for all comparisons). When the middle leg was the only leg of the tripod on the outside of the turn, it produced the greatest ΔP_leg in the largest percentage of steps (Kruskal–Wallis test, P=0.001). When the front and hind legs were the legs of the tripod on the outside of the turn, the hind leg produced the greatest ΔP_leg in a larger percentage of steps than the front leg (Kruskal–Wallis test, P<0.001).

One leg produced a majority of the total force impulse responsible for changing the heading of the animal. When a leg produced the largest ΔP_leg, the impulse generated by that leg averaged 79 % of the total perpendicular force impulse in the trial turn direction. For the outside legs, this impulse was significantly greater than the 33 % that would be expected if all the legs contributed equally to deflecting the body (t-tests, P<0.001 in all cases; Table 4). Inside front legs produced impulses that were significantly different from 33 % of the total perpendicular force impulse (t-tests, P<0.05). Inside middle and inside hind legs produced force impulses that were not significantly different from 33 % of the total because of the low sample sizes for these legs.

Legs most often produced torque that tended to rotate the body away from themselves (Table 4). Legs on the outside of the turn tended to produce positive torque rotating the body towards the trial turn direction (Fig. 6). Legs on the inside of the turn tended to produce negative torque rotating the body against the trial turn direction. The outside middle leg generated ΔL_leg towards the trial turn direction significantly more frequently than did the outside hind leg (Cochran Q-test, P<0.05), but not significantly more frequently than the outside front leg (Cochran Q-test, P>0.7).

When the middle leg was the only leg of the tripod on the outside of the turn, it produced the greatest ΔL_leg towards the trial turn direction in the largest percentage of steps (Kruskal–Wallis test, P<0.001). When the front and hind legs were the legs of the tripod on the outside of the turn, they produced the greatest ΔL_leg towards the turn with similar frequency, but in a significantly greater proportion of steps, than the inside middle leg (Kruskal–Wallis test, P=0.016).

When a leg produced the largest ΔL_leg, the impulse generated by the leg averaged 83 % of the total torque impulse in the trial turn direction. This indicates that one leg produced a majority of the total torque impulse responsible for rotating the animal. For the outside legs and the inside front leg, this impulse was significantly greater than the value of 33 % that would be expected if all legs were contributing equally to deflecting the body (t-tests, P<0.001 in all cases). The inside middle and inside hind leg produced torque impulses that were not significantly different from 33 % of the total because of the low sample sizes for these legs.

Moment arms contributing to torque production for the outside front and middle legs were significantly greater than moment arms for the outside hind legs (ANOVA, P<0.001 for both comparisons) but were not significantly different from each other (ANOVA, P=0.98; Table 4). Moment arms for the inside middle leg were significantly greater than those for the front and hind legs on the inside of the turn (ANOVA, P=0.013 and P=0.005, respectively), but moment arms for the front and hind legs were not significantly different from each other (ANOVA, P>0.9).

Turns during rapid running were unsteady and dynamic. Turns were characterized by periods of straight-ahead running punctuated by periods where the heading was deflected, resulting in movements of high curvature. The speed of the COM was not constant over the course of a single turn trial (Fig. 4B). Cockroaches accelerated and decelerated within each step (Fig. 4C), similar to the mechanics observed during constant average-speed, straight-ahead running (Full and Tu, 1990).

Curvature fluctuated during turning for rapid-running cockroaches (Fig. 4D), resulting in θ_d values both towards and against the trial turn direction (Table 2). Cockroaches were capable of turns with curvatures comparable with those reported for ants Cataglyphis albicans (typical values of 0.25 mm⁻¹, maximum of 1.5 mm⁻¹; Zollikofer, 1994), fruit flies Drosophila melanogaster (maximum of 0.1 mm⁻¹; Götz and Wenking, 1973) and stick insects Carasius morosus (0.1–0.2 mm⁻¹; Jander, 1985) moving at approximately one-tenth of the speeds used by the cockroaches in the present study (Table 2).

In contrast to the changes in heading deflection observed during turning, cockroaches maintained their rotational velocity in the trial turn direction (Fig. 4F). Only three out of 41 steps exhibited net rotational velocity against the trial turn direction (Table 2). The rotational velocities measured during turning were an order of magnitude greater than the fluctuations in rotational velocity reported for crayfish (Procambarus clarkii) turning at 6 cm s⁻¹ (Domenici et al., 1998). Unlike cockroaches, the rotational velocities observed in crayfish fluctuated around zero. Rotational velocity in cockroaches fluctuated within steps due to ΔL both towards and against the trial turn direction. The observed torque impulses were due to large fluctuations of rotational acceleration throughout turns (Fig. 4G). However, ΔL against the trial turn direction was similar to ΔL in the trial turn direction, causing a positive rotational velocity to be maintained throughout the turn. Rotational acceleration also alternates direction within steps during straight-ahead locomotion (Full and Tu, 1990; Kram et al., 1997), but rotational velocity fluctuates around a value of zero.

Cockroaches performed turns of magnitudes similar to those of other insects moving at approximately one-tenth of the speeds used in the present study. The magnitudes of fore–aft axis rotation that we measured for turning cockroaches (5–20 ° per step; Table 2) were comparable with the rotation magnitudes measured in the same species during slow running (15–30 ° per step; Watson and Ritzmann, 1998) and those measured in stick insects moving at approximately 0.5–2 cm s⁻¹ (10–20 ° per step; Jander, 1985). More rapid turns have been described in American cockroaches (Periplaneta americana) during escape turning. Periplaneta americana can rotate by 90 ° initially over the course of one stride period (Camhi and Levy, 1988) and can execute alternating turns at frequencies of 25 Hz (Camhi and Johnson, 1999). The turns we measured can be considered as turns of moderate magnitude and are most certainly not the maximum performance of these animals.

Despite the dynamic nature of turning, cockroaches maintained their fore–aft axis close to the direction of their heading (Fig. 4E). The magnitude of heading deflection for the entire trial (48 °) matched the magnitude of fore–aft axis rotation (40 °). The mean value of Δθ throughout the turns was only 5 ° (Table 2).

Cockroaches decelerated and accelerated both in speed and in rotational velocity during turning, typical of the bouncing gait observed during straight-ahead running (Full and Tu, 1990; Kram et al., 1997). Minor alterations in the perpendicular force impulses and rotational torque impulses produced by cockroaches during straight-ahead running could account for the whole-body mechanics observed during turning (Fig. 7). Heading deflection resulted from a decrease in ΔP relative to straight-ahead running by the tripod containing the outside middle leg and from instances of ΔP production in the direction opposite to that observed during straight-ahead running (Table 3). Both effects could result from an increase in force production or stiffness of the outside middle leg. Body rotation resulted from both tripods deviating from straight-ahead running by producing non-zero values of ΔL during a step.

The tripod containing the inside middle leg was responsible for the majority of ΔP and ΔL production in the trial turn direction. This tripod produced ΔP in the trial turn direction in 18 steps and ΔL in the trial turn direction in 12 steps of the 21 steps observed (Table 3). During turning, this tripod produced ΔP in the same direction and of approximately equal magnitude to the ΔP expected during straight-ahead running. As for straight-ahead running, ΔP was directed towards the middle leg in a majority of steps (Fig. 7). Even though the tripod containing the inside middle leg generated the largest ΔP in the trial turn direction, this ΔP was only 25 % greater than the ΔP observed during straight-ahead running. The force impulses observed during straight-ahead running are sufficient to generate turns of magnitudes comparable with those measured in the present study.

The tripod containing the outside middle leg generated ΔP in the trial turn direction (in the direction away from the middle leg) in six out of 20 steps (Table 3). The tripod containing the outside middle leg thus also contributed to deflecting the heading in the trial turn direction. The direction of ΔP production during these steps represents a departure from the ΔP exhibited during straight running, which is directed towards the middle leg (Full et al., 1991; Full and Tu, 1990). The tripod containing the outside middle leg produced ΔP in the trial turn direction of approximately half the magnitude of the alternative tripod. Force impulses produced by this tripod against the trial turn direction were also nearly half those expected during straight-ahead running. The shift in ΔP production by the tripod containing the outside middle leg could result from a decrease in force production by the inside front or hind legs, from an increase in force production by the outside middle leg, or from both.

The tripod containing the inside middle leg generated ΔL in the trial turn direction in a majority of steps (Table 3). This resulted from the fact that both tripods generated torque impulses in the direction of their middle leg more frequently than in the opposite direction. The magnitudes of the torque impulses within single steps during turning were greater than those during straight-ahead running (Fig. 7). During straight-ahead running, values of ΔL over single steps are close to zero since rotational velocity reaches zero at step transitions (Kram et al., 1997). Torque towards the middle leg in the first half of the step causes rotation towards the middle leg, but the rotational acceleration caused by this torque is cancelled by torque against the direction of the middle leg in the second half of the step. The opposing torques within each step during straight-ahead running result in zero net torque impulses over the entire step.

During turning, torque impulses in both directions did not oppose each other equally within individual steps, resulting in both tripods producing ΔL towards and against the trial turn direction. Non-zero torque impulses by both tripods may act to maintain a positive rotational velocity over the duration of the turn (Fig. 4F; Table 2). As for straight-ahead running, in a majority of steps during turning, both tripods generated torque impulses that would act to rotate the fore–aft axis towards the middle leg if the rotational velocity were zero at the beginning of the steps (Table 3).

The stride frequencies, duty factors and leg phases used during turning were similar to those during straight-ahead running at the same average speed. The mean stride frequency measured during turning of 7.3 Hz is similar to those reported for the same species during straight-ahead running (8–10 Hz; Full and Tu, 1990; Ting et al., 1994). Duty factors during turning averaged 0.56, similar to those reported for straight-ahead locomotion in Blaberus discoidalis (0.6; Ting et al., 1994).

The phase relationships found in cockroaches during turning show that cockroaches maintained a strict alternating tripod gait (Table 1), as observed during rapid straight-ahead running. Cockroaches did not turn by altering stride frequency on one side of the body relative to the other side. This is not surprising considering that cockroaches would have to nearly double their stride frequency on one side of the body relative to the other to produce the turns measured in the present study. Turning without altering stride frequency is common among arthropods moving at a range of speeds. Fruit flies Drosophila melanogaster (Strauß and Heisenberg, 1990), 12 species of ants (Zollikofer, 1994) and beetles Geotrupes stercorosus (Frantsevich and Mokrushov, 1980) have also been found to turn during walking without altering their stride frequency. Carausius morosus nymphs (Graham, 1972) and adults in gradual turns (Jander, 1985) also maintain tripod gaits during turning. Similarly, tethered crayfish (Astacus leptodactylus) (Cruse and Silva Saavedra, 1996), stationary cockroaches Blattella germanica (Franklin et al., 1981) and spiders Metaphidippus hartfordi (Land, 1972) turn without altering the step frequency on one side of the body. Cockroaches rotating in place have a tendency towards tripod coordination (Franklin et al., 1981). Strict tripod coordination might be expected in rapid locomotion, where centrally generated, feedforward commands have been predicted to become more important in locomotory control (Pearson, 1972).

Foot placement and leg movements of turning cockroaches could not be related unambiguously to leg force or torque production. Kinematic studies of maneuvers in insects frequently infer the patterns of force production and the contributions of legs to accelerating the body from the movements of the legs relative to the body in free or constrained conditions. This has led to the hypothesis that all the legs contribute to escape turning (Camhi and Levy, 1988; Schaefer et al., 1994; Tauber and Camhi, 1995). The assumption that all the legs can contribute to maneuvers is supported by the finding that any of the legs can contribute to turning the body if other legs are removed (Götz and Wenking, 1973). However, leg movements do not necessarily reflect the magnitudes of forces produced by the legs, and animals can execute turns that appear kinematically similar yet require very different amounts of force and torque production (Land, 1972). In addition, when turning during forward running, it has been noted that leg movements can result from the movement of the body over the legs and not from active force generation by the legs themselves (Domenici et al., 1998). In contrast to the movements of the feet relative to the body during the stance period, the placement of the feet at touch-down (r_AEP and ϕ_AEP) relative to the body does not depend on the movements of the body over the substratum. Changes in foot placement could therefore indicate active alterations of leg kinematics involved in turning (Cruse and Silva Saavedra, 1996). For example, Franklin et al. (1981) found that pure rotation involved measurable changes in AEP angles.

During rapid running in cockroaches, turning resulted from changes in force production among the legs that were not related to measurable differences in foot placement kinematics or in movements of the legs relative to the body during stance. The orientation of the legs at the AEP (ϕ_AEP) did not differ from scaled values for straight-ahead running (Fig. 5). Functional lengths (r_AEP) differed from straight-ahead running for only two legs, one of which (the outside front leg) contributed to turning the body and one of which (the inside hind leg) did not. Most of the legs showed no significant differences from straight-ahead running or from the contralateral leg, whether or not the leg contributed to turning the body. This finding is similar to measurements on crayfish (Astacus leptodactylus), which also did not alter their AEP positions during turns (Cruse and Silva Saavedra, 1996). Consequently, kinematic changes in leg placement at the AEP did not uniquely determine leg force or torque production during turning.

Differences in ϕ_PEP and r_PEP values between legs indicate that leg excursions did differ from values for straight running and differed between contralateral legs during turning (Fig. 5). Differences in leg excursions between contralateral legs were consistent with the requirements for altering stride length on the outside of the body imposed when turning at a constant stride frequency and duty factor. Large differences in r_PEP relative to straight running for the inside front and middle legs might suggest that these legs were actively involved in turning. However, the inside legs actively pushed against the trial turn direction in the majority of steps (Table 4). In addition, measurable force production by individual legs did not always last for the duration of the stance period (Fig. 6), further complicating the relationship between stride kinematics and leg force production. Legs can remain in contact with the substratum and undergo large excursions but still only produce force for part of the stance period.

Since three legs contact the substratum at any one time, the individual leg forces that cause the ΔP and ΔL that deflect the heading and rotate the body are indeterminate. Estimating force and torque production using kinematic data was not possible since kinematics do not uniquely determine force or torque production.

Single-leg force measurements showed that, during turning, F_p for all legs was directed towards the fore–aft axis and heading axis. The inside legs pushed outwards, against the trial turn direction (Table 4). Consequently, legs on the outside of the turn generated the force and torque impulses necessary to turn the body.

In many steps, the outside front legs contributed substantially to deflecting and rotating the body, resulting in large deflecting forces and rotational torques (Fig. 6). The outside front legs generated the greatest ΔP_leg and ΔL_leg in approximately half the trials (Table 4). Low force production relative to the hind legs in some steps prevented the outside front legs from generating the greatest ΔP_leg in the trial turn direction. In spite of low force production relative to the hind legs, the outside front legs produced the greatest ΔL_leg in the trial turn direction in a majority of steps (Table 4).

The outside middle legs generated the greatest ΔP_leg and ΔL_leg in the trial turn direction in approximately 80 % of the steps of this tripod. The tripod containing the outside middle leg generated ΔP and ΔL in the trial turn direction with lower frequency than the alternative tripod (Table 3). However, the action of the outside middle leg may have been important in reducing the ΔP and ΔL produced by this tripod and in causing ΔP to be directed in the trial turn direction in 30 % of steps and ΔL to be directed in the trial turn direction in 40 % of steps.

The shift in ΔP production by the tripod containing the outside middle leg could be due to a decrease in force production by the inside front or hind legs, to an increase in force production by the outside middle leg, or to both. Our data are consistent with the findings of Watson and Ritzmann (1998), which show electromyographic frequencies for outside middle legs that are greater than values recorded during straight-ahead running.

Outside hind legs generated the greatest ΔP_leg among the legs of the tripod in a majority of steps because the large forces had large components perpendicular to the heading (Table 4). Despite high force production, the hind legs did not generate the greatest ΔL_leg in a majority of steps. The large forces produced by the hind legs were associated with small moment arms (approximately 1 mm; Table 4), indicating that the forces were, on average, directed towards the COM. This resulted in small torques about the COM.

Interpretation of the complex dynamics of turning requires a simple horizontal plane model. Our model assumes that three legs of a tripod act as a single leg. Using this simple model allows patterns of leg force production to be related to the morphology of the animal (i.e. the body shape and leg position). The efficacy of a single leg in deflecting and rotating the body can be evaluated as a function of its position.

The cockroaches in the present study were able to generate perpendicular force impulses comparable to three-quarters their linear momentum, as indicated by their LMN of 0.75 (Table 2). For comparison, a circular track would need to have a radius of curvature (R) of 3–4 m to require similar performance from a 60 kg human running at 5 m s⁻¹ with a stance period of 500 ms. This would represent near-maximal performance for a human (Greene, 1985).

If the animal moves along a circular curve, the force relative to the direction of motion (F_p, the centripetal force) required to deflect the COM has a constant magnitude, mV²/R. Other patterns of force production, which do not result in curved trajectories, are also possible.

A second requirement for a successful turn involves rotating the fore–aft axis so that it aligns with the direction of the velocity vector (Fig. 1B). Minimizing the degree of misalignment (Δθ) between θ_d and θ_r is important for resuming straight running following a turn. To rotate its body, the animal must generate a net torque (T) about the COM. Assuming that the rotational velocity (ω) is zero at the beginning of the turn, to minimize the net Δθ at the end of the turn, the double integral of net angular acceleration (i.e. T divided by the yaw moment of inertia, I) with respect to time over a step period must equal θ_d. In the simplest case of a constant net torque over time, the body will have a positive ω at the end of the turn, which must be arrested in subsequent steps. We will not consider other possible cases, such as using time-varying net torque production or constraining foot placement to maintain ω=0 at the end of the turn.

Many-legged terrestrial animals must generate net forces and torques on the COM by pushing or pulling against the substratum with their legs. A leg used to generate the perpendicular force impulse necessary to change the heading of the body may not be in an advantageous position to simultaneously generate a torque impulse to rotate the body in the appropriate direction. For example, if a leg located behind the COM (e.g. the hind legs in cockroaches) exerted a purely perpendicular force to deflect the heading, the torque resulting from this force would cause the body to rotate in the direction opposite to the direction of deflection, causing the fore–aft axis and heading to become misaligned.

The present model assumes that animals produce forces perpendicular to the heading necessary to deflect the body using one leg initially placed at a set distance from the COM. The model allows calculation of the forces parallel to the heading necessary to rotate the body to keep the fore–aft axis aligned with the heading at the end of the step period, τ. Important simplifying assumptions of the model are as follows: (1) the heading and fore–aft axes are initially coincident with the x-axis of the global reference frame; (2) the animal’s COM moves around a circular trajectory by generating a constant force perpendicular to the heading (F_p) using one leg placed at an initial distance in the initial heading direction (P_AEP,ih) from the COM; (3) the animal maintains an approximately constant speed (V) over the course of the turn; (4) the leg moves approximately parallel to the heading over the course of the turn; and (5) the turn begins with zero angular velocity, ω.

Equation 6 indicates that for a given value of ε, the F_h necessary to prevent under-rotation or over-rotation of the body is directly proportional to the placement of the leg perpendicular to the heading. The leg placement determines the amount of force parallel to the heading necessary to minimize net Δθ. Fig. 8 shows that, for cockroaches, as the initial leg placement moves farther from the fore–aft axis (as |P_p| increases), the model predicts that the required F_h decreases. The resultant force vectors change direction to become closer to being perpendicular to the heading.

The total force predicted to turn the body is a nonlinear function of P_AEP,ih and P_p (Fig. 8). The predicted total forces for the outside legs required to turn the body (6–15 mN) are similar to the total horizontal forces predicted for single legs of the animal during straight-ahead running (8–10 mN; scaled from single-leg forces reported in Full et al., 1991, to take the larger mass of the animals in the present study into account). However, the forces required are approximately one-tenth the maximum single-leg forces that these animals are able to produce (over 100 mN; Full and Ahn, 1995; Full et al., 1995). Equations 6 and 7 suggest that all legs could contribute to deflecting and rotating the body. Table 4 shows that front legs are predicted to operate at an ε close to 1, indicating that the F_p required to deflect the heading should also be appropriate to rotate the fore–aft axis to minimize net Δθ. The middle and hind legs could also deflect and rotate the body to turn, but these legs operate at ε<1 and, consequently, must generate larger F_h and thus larger total forces.

The model expressed by equations 5–7 predicts leg force production for the outside legs. At an initial ω of zero, the model predicts that the front legs should produce almost purely perpendicular forces (ε close to 1; Table 4) and that the middle and hind legs should generate acceleratory forces (ε<1; Table 4).

The outside front legs operate at an ε of approximately 1 (Table 4). The front legs are predicted to be effective at rotating the body simply by generating the perpendicular forces necessary to deflect the body. The force angle predicted from the ε of the front legs is 93 ° (assuming a 5 g animal turning in 60 ms with the average initial foot placements shown in Fig. 8). The mean direction of force production measured during turning for the front legs was 87±15 ° (mean ± S.E.M.; Table 4), indicating that cockroaches turned by generating nearly perpendicular forces with their front legs.

The mean direction of force production for the outside middle legs was 38±13 ° (mean ± S.E.M.; Table 4). The middle legs generated large forces parallel to the heading, causing the force angle to decrease relative to the front legs. The middle legs operate at an ε of 0.48, indicating that the perpendicular forces necessary to deflect the heading rotate the fore–aft axis in the correct direction, but do not result in adequate rotation to maintain the fore–aft axis aligned with the heading at the end of the turn. The force direction predicted for the middle legs, given an ε of 0.48, is 51 °, slightly higher than the observed force direction.

The mean direction of force production for the outside hind legs was 45±14 ° (mean ± S.E.M.; Table 4). The hind legs also generated large forces parallel to the heading, causing the force angle to decrease relative to the front legs. This was due to the fact that the hind legs operate at an ε of approximately −0.4, indicating that the perpendicular forces produced by the hind legs act to rotate the body against the turn direction. The force direction predicted for the hind legs, given an ε of −0.42, is 22 °, slightly lower than the observed force direction.

The front legs produced forces nearly perpendicular to the heading direction, and the middle and hind legs produced additional forces in the heading direction. These changes in force direction are consistent with the predictions of the turning model. The moment of inertia of cockroaches is large enough relative to their body mass to require that the front legs produce almost completely perpendicular forces, and that the middle and hind legs produce large acceleratory forces, to maintain the fore–aft axis aligned with the heading at the end of the turn. The force directions predicted by the model and observed in cockroaches could result from minor alterations in the forces produced during straight-ahead running. During straight-ahead running, the front, middle and hind legs have mean directions of ±134 °, ±90 ° and ±33 °, respectively (force directions calculated from force maxima reported in Full et al., 1991). The directions of force production by the inside legs (−119 ° for the front legs, −76 ° for the middle legs and −41 ° for the hind legs; Table 4) are closer to the values observed during straight-ahead running than the values for the outside legs. Although we cannot determine whether the inside legs change the magnitudes of their force production, the direction of force production by the inside legs remained similar to the force directions during straight-ahead running. This suggests that turning resulted from changes in the direction of force production by the outside legs.

Lateral force impulses observed during straight-ahead running are sufficient to generate the perpendicular force impulses observed during turning. However, the torque impulses observed during straight-ahead running do not generate rotations sufficient to maintain the fore–aft axis aligned with the heading. To generate additional fore–aft axis rotation, cockroaches could simply reduce the deceleratory forces parallel to the heading direction produced by the front legs and increase the acceleratory forces parallel to the heading direction produced by the middle and hind legs. As a result, force direction angles for all legs should decrease relative to straight-ahead running. This is consistent with the force direction angles observed for the front and middle legs during turning, which are, respectively, 47 ° and 52 ° smaller than those observed during straight-ahead running. The hind legs, which did not generate the majority of ΔL, showed mean force direction angles 12 ° greater than those observed during straight-ahead running. However, the mean force direction angle during straight-ahead running was only 11 ° greater than the force direction predicted from the leg effectiveness number of the hind legs during turning. The direction of force produced by the hind legs was only 23 ° greater than the force direction predicted to minimize net Δθ.

The line containing the predicted resultant force vector intersects the fore–aft axis at a point forward of the center of mass (Fig. 8). This point is analogous to the attachment point of a single effective ‘leg spring’ that can model the dynamics of hexapodal locomotion in the horizontal plane (J. Schmitt and P. Holmes, in preparation). Turning with this single effective ‘leg spring’ is greatly facilitated by moving the attachment point forward of the center of mass, similar to the prediction of the present model.

Our study of maneuverability in cockroaches has shown that a many-legged, sprawled-posture morphology provides a range of immediately available options for maneuvering. The front, middle and hind legs on either side of the body could contribute to turning the body. The front legs operate at a leg effectiveness number of approximately 1, indicating that they can contribute effectively to turning by producing forces nearly perpendicular to the heading. The middle and hind legs operate at leg effectiveness numbers of less than 1, indicating that they must produce additional forces parallel to the heading, which accelerate or decelerate the animal.

The cockroaches used only a subset of the options available to them for deflecting and rotating the body. During rapid running, they used only their outside legs to generate the perpendicular forces and rotational torques necessary to deflect their heading and rotate their fore–aft axis in the trial turn direction. The inside legs acted against the turn, constraining the turning performance.

The perpendicular force impulses observed during straight-ahead running are sufficient to generate turns of the magnitudes we observed when unarrested by subsequent steps. Moreover, the single-leg ground-reaction forces predicted to turn the body based on the leg effectiveness number and the single-leg forces actually observed during turning were similar to the patterns of leg force production observed during straight-ahead running. Consequently, our analysis has revealed that the patterns of force production observed in these sprawled-postured hexapods during straight-ahead running provide the possibility for remarkable maneuverability, while requiring only minor adjustments in leg force production. The morphological design of these organisms (i.e. the leg number, leg position, mass and moment of inertia), coupled with their locomotion dynamics during straight-ahead running, could considerably simplify the control of maneuvers such as turning.

The mechanical consequences of using the front, middle or hind legs to turn have potential applications to simplifying the control of maneuvers under different conditions. The front legs are in an advantageous position to turn the body while minimizing net Δθ when initiating a turn from a rotational velocity close to zero. In addition, the front legs can contribute to turns without accelerating or decelerating the body. The middle and hind legs are in advantageous positions to turn the body while minimizing net Δθ from initially positive rotational velocities or when acceleration is advantageous. Whether animals utilize their morphological design to simplify the execution of an array of different maneuvers under different conditions is a behavioral question that deserves future study.

The leg effectiveness number that we present depends on the assumptions that the fore–aft axis and the heading remain parallel, and that the movement of the foot relative to the body and heading during a turn is approximately linear (solely in the fore–aft or heading direction). These assumptions are reasonable given the small (approximately 5 °) mean difference measured between the fore–aft axis and heading (Table 2) and that the motions of the outside legs are approximately parallel to the fore–aft axis (Fig. 5). Foot positions and forces expressed relative to the heading axis are thus assumed to equal foot positions and forces expressed relative to the fore–aft axis. Movements of the COM and rotation of the body are assumed not to cause the foot position to deviate substantially from a linear trajectory. P_h decreases with constant velocity, and P_p remains approximately constant. Consequently, we would expect ε to describe only turns of relatively small magnitude. Simulations with a less simplified model show that errors in the resulting deflection and rotation from forces calculated using ε are expected to be small for leg position values in the neighborhood of those used by the cockroach and for the mean rotations per step observed in the present study.

During steady-speed, straight-ahead running, rotational velocity reaches a minimum at step transitions (Kram et al., 1997), so the assumption of ω=0 at the beginning of a turn is reasonable. During turning, rotational velocity was directed towards the trial turn direction for the majority of steps (Table 2), which would cause the torque impulses due to forces along the heading axis necessary to rotate the body to decrease, thus decreasing the predicted forces along the heading axis required of the legs. However, since the contribution of angular velocity to rotation is likely to be different for the first and subsequent steps, and is unlikely greatly to change the predictions of the model, we considered only the case of initially zero rotational velocity.

 
• a_I–a_IV, a_IV*

  areas of quadrants I–IV (see Fig. 7 in Full et al., 1995)

 
• AEP, PEP

  anterior, posterior extreme position

 
• C, K, K₁, K₂

  constants

 
• COM

  center of mass

 
• F

  resultant horizontal force

 
• F_h component of F

  in the direction parallel to the heading

 
• F_p component of F

  in the direction perpendicular to the heading

 
• I

  moment of inertia about the vertical (yaw) axis

 
• l

  moment arm

 
• LMN

  linear maneuverability number

 
• m

  body mass

 
• M₁

  slope of the relationship between the quadrant span and σ_h (see Full et al., 1995)

 
• M₂

  slope of the relationship between the quadrant ratio and σ_h/σ_t (see Full et al., 1995)

 
• P_AEP,ih

  initial location of the foot in the initial heading direction at the beginning of the step

 
• P_h

  location of a foot relative to the center of mass in the direction parallel to the heading

 
• P_p

  location of foot relative to the center of mass in the direction perpendicular to the heading

 
• r

  distance from the center of mass to a foot

 
• r_AEP

  distance from the center of mass to a foot at the AEP

 
• r_PEP

  distance from the center of mass to a foot at the PEP

 
• R

  radius of curvature

 
• t

  time

 
• V

  speed of the center of mass

 
• ε

  leg effectiveness number

 
• ΔL

  torque impulse

 
• ΔL_leg

  torque impulse generated by an individual leg

 
• ΔP

  force impulse in the direction perpendicular to the initial heading

 
• ΔP_leg

  force impulse in the direction perpendicular to the initial heading generated by an individual leg

 
• Δθ

  degree of misalignment between the fore–aft axis and the heading (Δθ=θ_r−θ_d)

 
• ϕ

  angle between the fore–aft axis and a foot

 
• ϕ_AEP

  angle between the fore–aft axis and a foot at the AEP

 
• ϕ_PEP

  angle between the fore–aft axis and a foot at the PEP

 
• θ_d

  angular change of heading (magnitude of deflection) relative to the global coordinate frame

 
• θ_r

  angular change of the fore–aft axis (magnitude of rotation) relative to the global coordinate frame

 
• θ_p

  change in orientation of the fore–aft axis relative to the coordinate frame set by the initial fore–aft axis due to a force perpendicular to the heading

 
• θ_h

  change in orientation of the fore–aft axis relative to the coordinate frame set by the initial fore–aft axis due to a force parallel to the heading

 
• σ_h

  horizontal stress

 
• σ_t

  total stress

 
• τ

  step period

 
• T

  torque about the center of mass

 
• T_p

  torque about the center of mass caused by F_p

 
• T_h

  torque about the center of mass caused by F_h

 
• ω

  rotational velocity

This work was supported by an NSF Graduate Research Fellowship to D.L.J., ONR Grant N00014-92-J-1250 and DARPA Grant N00014-98-I-0747 to R.J.F. We would like to acknowledge Tim Kubow for his insights and contributions, and James Glasheen for deriving the linear maneuverability number. We would also like to acknowledge Jeremy Berger and John Chung for assisting in data collection and analysis, and Kellar Autumn for statistical advice. We would also like to acknowledge Andre Seyfarth, Rodger Kram and the Berkeley Biomechanics group for critically reading the manuscript.