We designed an automated data acquisition system that generates a large positional dataset with high spatial resolution to investigate the fly’s gait and CoM kinematics. Similar to an approach employed previously (Nye and Ritzmann, 1992; Wosnitza et al., 2013), we recorded the side and the bottom (reflected off a mirror) view of a fly walking in a clear, closed cuboid chamber (Figure 2A). We extracted all the steps during which a fly walked straight for more than one step. The fly’s CoM was extracted using the Kanade–Lucas–Tomasi (KLT) (Tomasi and Kanade, 1991) algorithm and produced low-noise estimates of the CoM position; the vertical resolution being 20 μm (see 'Materials and methods', 'Tracking CoM and foothold positions', Video 1), which makes the rhythmic up-and-down movement of the CoM apparent (Figure 2B). The positions of the leg tips during stance were extracted using a custom algorithm (see 'Materials and methods', 'Tracking CoM and foothold positions', Figure 2—figure supplement 1). The legs were labeled according to an established convention (Figure 2C), and the gait map (Figure 2C) was put together such that the legs that constitute a tripod – right prothoracic (R1), left mesothoracic (L2), and right metathoracic (R3) – are plotted on consecutive rows (orange); and those of the other tripod (L1–R2–L3) are plotted in another set of consecutive rows (marked in blue), to allow a direct assessment of the presence or absence of the tripod gait.

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Video 1

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As a means of corroborating previous findings, we plotted stance and swing duration as a function of speed (Figure 2D). Consistent with previous studies (Graham, 1972; Mendes et al., 2013; Pearson, 1976; Strauss and Heisenberg, 1990; Wilson, 1966), the stance duration is inversely proportional to speed. The swing duration also changes with speed but to a smaller extent than the changes in stance duration.

We used two methods to characterize the speed-dependent change in coordination between legs: first, to facilitate comparison with previous work, interleg coordination was defined based on delays between the times at which the legs start either a swing or a stance phase. To visualize a fly’s gait, the times that a fly’s legs start the stance phase in relation to the time that the right front leg (R1) entered the stance phase were plotted. Legs that form the first set (R1–L2–R3) of tripod legs enter stance phase with a short interleg delay (Figure 3A). The legs that form the other tripod (L1–R2–L3) enter the stance phase with a short interleg delay with each other but out of phase with the first set. The coordination pattern did not change noticeably as a function of speed (Figure 3A). This raw gait map (Figure 3A) suggests that the flies predominantly employ a tripod gait across all speeds. This trend (in Figure 3A) was quantified by calculating the delays relative to the cycle period (the time it takes a leg to complete both a swing and a stance, Figure 3B) or normalized delay. The normalized delays between the legs of the same tripod were small throughout the entire speed range; the within tripod delays became even smaller with speed. The prothoracic leg led the other legs in its tripod with a small but significant negative delay consistent with previous observations in cockroaches (Bender et al., 2011; Delcomyn, 1971). On the other hand, the normalized delays across legs in the opposing tripods were 0.5 (Figure 3B). These analyses suggest that the gait – as defined by phase differences between legs – employed by flies during forward walking across the entire range of speeds is close to a tripod.

Figure 3

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The normalized delays between different legs are consistent with that of a single gait that is close to a tripod but in which the front leg of the tripod is ahead of the middle leg, which in turn is ahead of the rear leg; we will refer to this gait as M-tripod (Figure 3C). The delays between legs within a tripod do decrease slightly with speed (Figure 3C). The small dependence means that there is no qualitative change in gait. This small dependence on speed is consistent with the continuum of coordination patterns observed in a recent study (DeAngelis et al., 2019).

A second method to quantify leg coordination is to use instantaneous phase lags between legs (Figure 3D) averaged over a gait cycle (Couzin-Fuchs et al., 2015; Revzen and Guckenheimer, 2008). Although time delays are easier to visualize and phase lag more abstract, the latter provides a more accurate measure of coordination because it takes the entire step into account instead of only the beginning of stance (see 'Materials and methods' 'Gait analysis based on leg phases'). As in the case of stance start times, the distribution of phase lags between the reference leg (R1) and the other legs show small phase differences between the legs within a tripod and large phase differences between the legs in the opposing tripod (Figure 3E). The phase plots also reveal that the front leg of the tripod leads the middle and back legs. As the speed increases, the phase difference between the tripod leg decreases, and the spread of the phase difference also becomes smaller (Figure 3E). The analysis using instantaneous phases is consistent with a single gait – M-tripod – across the entire range of speed; the exact speed dependence of M-tripod gait using phase difference (Figure 3F) is slightly different from the speed dependence calculated from stance start times (Figure 3C).

A small percentage (about 4%) of steps at very low speeds did not conform to any gait, and a few steps had a tetrapod coordination pattern, but an overwhelming majority of the steps have a tripodal coordination. The rest of the study will focus on steps that have a tripodal coordination.

Given that flies can walk over their entire speed range using a M-tripod gait implies that a change in gait is not essential for a change in speed. To better understand the mechanism underlying change in speed, we focused on the tripodal steps and asked how the movement of the fly’s CoM over the tripod gait cycle changed with speed. Because the tripod legs are not perfectly in sync, we defined the tripod start as the halfway time point between the time that the very first foot of the current tripod lands and the last foot of the preceding tripod is lifted (Figure 4A). Similarly, we set the tripod end as the halfway point between the very first foot landing time of the following tripod and the last lift-off time of the tripod of interest (dotted blue lines in Figure 4A mark the start and end of each tripod). We will refer to the tripod stance as a step.

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Figure 4A shows the speed profile during a fast step. As previously reported for stick insects (Graham, 1985) and Drosophila (Graham, 1972; Mendes et al., 2013), the CoM typically reached a maximum horizontal speed at mid-stance (Figure 4A). Figure 4B shows the speed for a slower step. A slow step is characterized by both a lower initial speed and a smaller speed increase.

It is important to note that there is a mid-stance maximum in the height of the fly and that the flies are more erect when they are walking faster (Figure 4A, B). We will show later that this change in height is partially responsible for the increase in speed.

The change in speed within a step increased with the average speed during the step. (Figure 4C). At low walking speed, much of the change in speed was due to the increased initial speed. At higher walking speed, the mid-stance increases in speed made a greater contribution (Figure 4D).

The distance traveled over the tripod gait cycle also increases with speed (Figure 4E), and the duration of the tripod gait cycle decreases (Figure 4F). Thus, the increase in speed is due to both the greater distance travelled during the step, and faster steps. The longer and faster steps result both from a faster speed at the beginning of the step and a greater increase in speed during the step.

In the rest of the article, we will describe a simple mechanical model that not only describes the mid-stance maximum in speed during a step but also describes the changes in mechanics underlying changes in speed.

As described in the 'Introduction', an animal walking with a tripod coordination can be modeled as a springy tripod where a point mass is supported by three legs. For symmetry, in our model, these legs were of equal natural length, and the body’s movement within a step can be described as an arc about the middle leg. The body’s position at any instant is described by $\theta$, the angle the body makes with the vertical, and r is the length of the middle leg. The behavior of this mechanical system can be described by its elastic potential energy. As the body moves through its stance phase, this elastic potential energy changes as some legs stretch and others compress. The total elastic potential energy of a springy tripod (Figure 5) is simply the sum of the potential energies due to the three legs.

Figure 5

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where $R_{tri}$ is the natural length of the springy tripod; $r$ is the length of the middle leg; $\theta$ is the angle that it makes with the vertical axis, which are also identified with the radial and angular coordinate of ARSLIP; 2L is the spread of the tripod or the distance between the prothoracic and metathoracic legs of the tripod in the direction of walking; and k is the stiffness of each leg. The variables are also enumerated in Table 1 and shown in Figure 5.

We can show through a formal analysis using a Taylor series expansion of the Lagrangian for the springy tripod (see 'Materials and methods' 'Derivation of the formula relating tripod model to ARSLIP') that springy tripod approximately reduces to the ARSLIP model (Figure 5B). As a reminder, in the ARSLIP model, the three legs of the springy tripod are replaced by a single effective leg with a radial and an angular spring. Specifically, $V_{tri}$ is equivalent to the ARSLIP potential energy, V (Equation 2), for evolution that is close to the midpoint $(r_m$), and $\theta =0$.

The ${\left(r-R\right)}^2$ term corresponds to potential energy due to the radial spring aligned along the effective leg connecting the middle tripod leg to the CoM, and the ${\theta }^2$ term corresponds to the potential energy from an angular spring capturing the tangential restorative forces exerted by the front and back legs. In this model, the mechanics of an animal is controlled by the two spring constants, $k_s$ and $k_a$, which describe the stiffness of the radial and the angular spring, respectively, and the natural spring length, R.

In essence, the mechanics of the fly walking on a springy tripod can be described by the ARSLIP model. An important point that we will elaborate on later is that the springy tripod is a simplification of the actual configuration of the fly while it is walking, but the ARSLIP model is a more general model and can serve as an accurate model even without making the assumptions of the springy tripod.

The approach above is based on potential energy. The distinction between SLIP and ARSLIP is clearer when considering forces modeled in the two cases. In SLIP, forces are always along the leg. The ARSLIP model provides a mechanism by which tangential forces can also be transmitted to the body. Importantly, the angular spring forces switch direction at mid-stance, which means that they aid forward progression during the first half of the stance and oppose forward progression during the second half of the stance. This pattern is exactly opposite to the pattern created by SLIP. Depending on whether the leg spring dominates, or the angular spring, one can get a cockroach-like speed minimum or fly-like speed maximum at mid-stance.

We evaluated the performance of the SLIP and ARSLIP models by fitting them to the fly’s CoM kinematics. Because the stance times of two consecutive tripods can overlap substantially, a complete model would involve two effective legs, each of which functions as either SLIP or ARSLIP; this complete model with two effective legs would have too many parameters and might obscure many of the insights that we obtain from modeling. Therefore, we modeled the CoM kinematics of a tripod stance (as defined in Figure 4A) using a single effective leg. The model parameters approximate the control exerted by the fly at each step. In the ARSLIP model, the fly chooses as its initial condition the angle of attack ($\alpha$), angular speed ($\mathrm{\Omega }$), leg length ($r$), and radial speed ($\dot{r}$) at the beginning of the step. The evolution of the CoM depends on the angular spring constant ($k_a$), leg spring constant ($k_s$), and the natural leg length of the effective leg ($R$). The only difference between ARSLIP and SLIP is the absence of the angular spring, and hence there is no $k_a$ in SLIP.

We minimized the root mean squared error (RMSE) between the SLIP and ARSLIP-predicted position of the CoM, and the experimentally measured position using an optimization algorithm (see 'Materials and methods', 'System of ordinary differential equations for SLIP and ARSLIP and details regarding fitting ordinary differential equations to individual steps'). SLIP can model the small increase in the vertical position of the CoM, which results from two competing effects: height increase due to the progression of the CoM from its extremum to the vertical mid-stance position and a height decrease due to the compression of the leg spring (Figure 6A). However, as reported previously (Biswas et al., 2018), SLIP fails to describe the horizontal progression of the CoM. This failure is clear from a comparison of the experimental horizontal speed profile and the theoretical speed profile (Figure 6A, bottom panel). In contrast, ARSLIP can describe both the horizontal and vertical progression of the CoM (Figure 6B). In ARSLIP, the angular spring accelerates the CoM during the first half of the stance phase. It can, therefore, compensate for, or even overcome, the decelerating effects of the radial spring and gravity, and can model the mid-stance maximum in speed. ARSLIP presented significantly smaller RMSEs for both horizontal and vertical CoM displacements than SLIP (Figure 6C).

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That the ARSLIP model describes the CoM kinematics well means that two linear springs defined by their spring constants – $k_a$ and $k_s$– are sufficient to describe the fly’s CoM kinematics during a step. The range of parameter values for all the fitted steps in our dataset is shown in Figure 6—figure supplement 1. The median $k_s$ was 0.009 N/m. This spring constant implies that to support its mass of 1 mg or 10 μN weight, the fly compresses this effective spring by about 1 mm or approximately 50% of its length. During a step, the spring is always compressed such that its length is close to the fixed point of the spring (length at which the spring forces cancel gravitational forces) and oscillates about this fixed point without reaching its natural length. The magnitude of these oscillations is small and reaches a maximum of 10% (of its length at fixed point) about the fixed point. The nondimensional radial spring constant ${\gamma }_s$ (see 'Materials and methods' for definition) is ~2 compared to >10 for humans (Antoniak et al., 2019). The median $k_a$ was 1.1 × 10⁻⁸ Nm/radian. In nondimensional terms, the angular spring constant ${\gamma }_a$ was ~0.5, which is like the values obtained in humans. Therefore, compared to humans, the relative role of angular spring in flies is much larger.

Most animals increase their walking speed by decreasing the stance duration (Mendes et al., 2013; Pearson, 1976). Modeling stance using an effective limb that functions as a spring provides a simple explanation for the mechanical changes that accompany this decrease: in any two-dimensional motion – including walking in the sagittal plane discussed here – the vertical and horizontal motion must be synchronized by relating parameters governing the vertical and horizontal time scale. The vertical oscillatory motion is controlled by the radial spring constant, and the horizontal motion by the angular speed. As the walking speed increases and the stance duration decreases, the vertical oscillations must occur faster by making the effective leg stiffer (because the time needed for vertical oscillation decreases with increases in spring stiffness). This increase in stiffness has indeed been observed in humans (Antoniak et al., 2019). We found a similar increase in stiffness in flies (Figure 6D). As the fly’s walking speed increases, ${\gamma }_s$, the nondimensionalized version of k_s increases. The nondimensional angular spring constant, ${\gamma }_a$, increases as well; this increase accounts for the greater within-step increases in speed observed at higher speed (see Figure 4C).

In the next section, we will show that the mechanism underlying the change in spring constant is surprisingly a result of the change in the geometry of the tripod with speed rather than the change in spring constant of the individual legs.

In bipedal walkers, the only mechanism for increasing the overall stiffness of the system is to increase the stiffness of each leg. In the case of polypedal walkers such as insects, including flies, the geometry of the tripod itself is a parameter that can be adjusted to alter the stiffness of the overall system. To test the extent to which the geometry of the tripod on a given step influences its kinematics on that step, we derived the equivalent ARSLIP model that displays the same dynamics as the springy tripod around its mid-stance position using the empirically obtained tripod geometry (determined by the tripod spread, $L,$ and mid-stance height, $r_m$, in Figure 7A) and mass of the fly (see 'Materials and methods' for details). Specifically, using the following equations:

Figure 7

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We can relate the ARSLIP spring constants – $k_a$ and $k_s$ and R (the natural length of the ARSLIP spring) to $k$, the spring constant of individual legs, R_tri, the natural length of individual legs, and the tripod geometry (determined by $L$ and $r_m$ in Figure 7A).

If the spring constant of individual legs remains the same, the springy tripod model predicts that ${\gamma }_a$ and ${\gamma }_s$(nondimensionalized versions of $k_a$ and $k_s$) will both vary over a twofold range due to the variation in the observed geometry of the tripod. Specifically, in the range of values observed in flies, as $L/r_m$ ratio decreases or the tripod becomes narrow and tall, both ${\gamma }_a$ and ${\gamma }_s$ increase (Figure 7B).

We can exploit this dependence of ${\gamma }_a$ and ${\gamma }_s$ on the tripod geometry to examine how well the change in tripod geometry from one step to the next predicts the best fit ${\gamma }_a$ and ${\gamma }_s$ values from Figure 6. To this end, we determined a single $k$ and R_tri for each fly, which best satisfies Equations 3–5 for all the steps fitted with the ARSLIP model for that fly. To compare across flies, we converted the $k_a$ and $k_s$ values to their nondimensionalized versions ${\gamma }_a$ and ${\gamma }_s.$ We found that, despite all the simplifying assumptions, the predicted ${\gamma }_a$ and ${\gamma }_s$ derived from the tripod geometry were close to the optimal ${\gamma }_a$ and ${\gamma }_s$ obtained from the best fit to the CoM kinematics (Figure 7C). The similarity between predicted and fitted spring constants is particularly significant because the prediction for all the steps of a given fly was made with a single parameter set while fits were optimized for each step representing a large decrease in the number of parameters. These results show that the tripod geometry plays a critical role in governing the spring constants.

The strong correlation between spring constants predicted from the geometry and those from optimization suggests that changes in tripod geometry are employed by the fly to change speed. Since ${\gamma }_a$ and ${\gamma }_s$ increase with speed (replotted in Figure 7D), we anticipate that the increase in speed is usually reflected as a change in $L/r_m$ ratio (Figure 7E), implying that the change in tripod geometry is an important mechanism for the control of speed during walking.

Flies appear to predominantly employ a gait close to a tripod gait – M-tripod – across their entire range of speeds. These results are consistent with other studies in flies (DeAngelis et al., 2019; Strauss and Heisenberg, 1990). Similar observations have been made during free walking in other insects such as cockroaches (Delcomyn, 1985; Hughes, 1952; Spirito and Mushrush, 1979), ants (Reinhardt and Blickhan, 2014; Wahl et al., 2015; Zollikofer, 1994), and locusts (Burns, 1973).

The M-tripod gait itself is not fixed but has a small dependence on speed. This small dependence on speed as well as the increase in duty factor as the speed decreases implies that the average number of legs on the ground at any given time will decrease with speed. This increase in the number of legs has been shown to be important for stability (Szczecinski et al., 2018).

The M-tripod gait only applies to forward walking at a fixed speed. The entire complement of gaits that the fly employs to turn, accelerate, and decelerate remains to be determined. It is important to note too that our findings do not imply that the flies are only capable of a fixed gait. There is evidence that flies change their gait upon amputation (Isakov et al., 2016). In our dataset as well, there is clear evidence for tetrapod gait; however, the fraction of steps during which flies adopt a tetrapod gait is very small.

An unexpected result was the extent to which the shape of the tripod formed by the three tripod legs – particularly the ratio of the height of the tripod to its anterior-posterior spread – can explain a fly’s CoM kinematics during a step. Previous studies have shown that neither the swing duration nor the swing amplitude (the distance that a leg travels during the swing) changes much as the fly’s walking speed changes (Mendes et al., 2013; Strauss and Heisenberg, 1990; Wosnitza et al., 2013), a result that is confirmed in this study. Much of the change in speed results from a decrease in stance duration (Mendes et al., 2013; Strauss and Heisenberg, 1990; Wosnitza et al., 2013), another result consistent with this study. In other words, increase in walking speed results from an increase in the angular speed of the body about its stance legs. This increase requires two elements: a neural element whereby increasing the drive into the central pattern generators would cause them to cycle faster, as has been demonstrated in stick insects (Büschges et al., 2008). A biomechanical element: moving faster also requires larger forces from the ground, and a mechanically stiffer system (in this case, the mechanical system consists of the fly and the legs that support the fly) would be able to transmit more forces from the ground to the body. There are two mechanisms by which the system can become mechanically stiffer. Either a fly could make each leg stiffer just as has been shown in humans (Antoniak et al., 2019; Kim and Park, 2011) or it could change the geometry of the tripod to make the overall system stiffer. The data in Figure 7 is consistent with the second idea that the changes in the geometry of the tripod are the dominant component by which flies control the stiffness of γ_a and γ_s and thereby change their walking speed. Changes in γ_s allow the fly to adjust the stiffness of its mechanical system to the stance duration; a stiffer γ_s means shorter time period of oscillation. Changing γ_s through changes in geometry would also change γ_a. We regard the changes in γ_a as an inevitable consequence of the changes in γ_s; nonetheless, the increase does provide a parsimonious explanation for the greater mid-stance maximum in speed observed when the fly walks faster.

To our knowledge, the control of speed through tripod geometry has never been explored in any insect. One reason for this deficiency is methodological. Researchers usually collate their data across steps, trials, and individuals. This process is bound to obscure any trends in tripod shape; analysis at the level of single steps is necessary, and trends within an individual must be compared. Another methodological issue is that the height of the animal during locomotion is rarely measured.

In flies, the $L/r_m$ ratio varies between 1 and 2. How about other insects? We could obtain a rough estimation of the $L/r_m$ ratio for a few insects by piecing together information from some manuscripts or by measuring these ratios from the figures in the papers: for three species of stick insects, the ratio ranges between 2 and 3 (Theunissen et al., 2015), wood ants have a $L/r_m$ ratio of closer to 1 (Reinhardt and Blickhan, 2014), and cockroaches have a $L/r_m$ ratio closer to 6 (Ting et al., 1994). This large variation in the $L/r_m$ ratio will have a large effect on the CoM kinematics within a step. Given a leg stiffness γ, at low $L/r_m$ ratio, that is, when the legs are almost vertical, γ_s is large in comparison to γ_a; therefore, the deceleration due to γ_s as the fly approaches mid-stance cannot be compensated by the acceleration due to γ_a. As the $L/r_m$ ratio increases, the effects due to γ_a and γ_s are comparable for a range of $L/r_m$ values. At very large $L/r_m$ ratios, γ_s again dominates. These ideas can be formalized by deriving conditions for which the gait is cockroach-like versus fly-like (Appendix): whether the gait is cockroach-like or fly-like depends on the interplay between γ_a and γ_s, which in turn depends on the stiffness of individual legs and the geometry of the tripod. For a given $L/r_m$, there is a leg stiffness γ above which the kinematics change from a cockroach-like gait to a fly-like gait (Figure 8A). As expected, the data points for each of the fitted steps for the fly lie above the function that demarcates the two kinematic types. Importantly, the $L/r_m$ ratio for flies and several other insects places them in a regime in which the leg stiffness required for a fly-like gait is at a minimum (Figure 8A). Indeed, ants, stick insects, and flies all have fly-like kinematics. On the other hand, cockroaches have $L/r_m$ values that predispose them toward a mid-stance minimum velocity profile.

Figure 8

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Another important insight from this analysis comes from the dependence of γ_s on $L/r_m$ ratio (Figure 8B). At the $L/r_m$ ratios observed in flies and other insects such as ants and stick insects, small changes in $L/r_m$ will produce a corresponding change in γ_s (Figure 8B). On the other hand, at large $L/r_m$ ratio, such as those employed by a cockroach, the γ_s is constant and does not depend on the $L/r_m$ ratio. This difference makes sense. At the large Fr numbers employed by cockroaches, the CoM kinematics is dominated by the angular speed with which the body moves about its leg (Antoniak et al., 2019), and it is rather insensitive to the magnitude of γ_s. On the other hand, and as demonstrated for flies in this study, at lower speeds the mechanics is dominated by the spring constants γ_a and γ_s.

In sum, the strong dependence of mechanics on the $L/r_m$ ratio or tripod geometry can explain both why tripod gait can be employed across a range of speeds observed among different insects and that a given insect can control its speed in part by changing the geometry of its tripod.

The finding that the SLIP model employed in mammalian locomotion is also adequate as a model for cockroach running led to the idea that SLIP is a general model for locomotion regardless of how many legs are on the ground. However, it has been well known before the recent studies in flies that CoM kinematics for other insects are dramatically different from cockroach (Graham, 1985; Reinhardt and Blickhan, 2014), and cannot be explained by the SLIP model. To our knowledge, there have not been many attempts at arriving at a single model that can serve as a conceptual model that explains the diverse kinematics observed across the insect world.

The development of a general model will be aided by two important insights in this study: first, we show that the geometry of the tripod plays an important role in determining the mechanics of the CoM. It appears that cockroaches do walk with a particularly wide and low tripod where the mechanics are closest to being described by the SLIP model. But the wide and low tripod observed for cockroaches appear to be an exception rather than a rule. The tripod geometries of other insects are different, and as shown in Figure 8, at least some of the kinematic variations observed in insects result from these differing tripod geometries.

Second, the effect of the different tripod geometries can be captured through a simple extension of the SLIP model – the ARSLIP model. The ARSLIP model is simple enough that it retains much of the elegant simplicity that makes SLIP such a powerful model while being better equipped to capture the greater diversity of kinematics observed in insects.

Future experiments aimed at exploiting the natural diversity of the insect world to rigorously test the relationship between the geometry of the stance legs and CoM kinematics, and the ability of ARSLIP to describe this relationship will prove to be a powerful toolkit for developing a general model for hexapedal locomotion.

It is obvious but still important to note that the springy tripod presented here is a simplification for the actual dynamics of locomotion in insects. In the case of fly, the springy tripod is a decent model of the fly’s walking as shown by its ability to predict the optimized ARSLIP spring constants from geometry (Figure 7). However, the ARSLIP model itself is more general and can model other known features of insect legged locomotion that we have not considered here:

• First, here we have modeled each leg as a linear spring, the angular spring results from a combined action of the three legs. In the most general case, each leg itself can function as both linear and angular spring. The resulting model will still be the ARSLIP model; however, the expressions relating $k_a$ and $k_s$ to tripod geometry and stiffness of a given leg will be different from what we have derived here.

• Second, and like the first point above, apart from spring forces, insect legs can produce attachment forces (Gorb et al., 2002). Once again, attachment forces do not invalidate the utility of ARSLIP as a model, but will affect the values of $k_a$ and $k_s$ differentially and represent an important mechanism that can explain the difference in kinematics for different insects.

• Third, for many insects, the hind leg is much longer than the other legs. The longer length of the hind legs might make a third term in the Taylor series expansion necessary (the first two terms being the leg spring and the angular spring, respectively). This third term may act as an asymmetric propulsive force.

• Finally, ARSLIP does not necessarily need three legs. The ARSLIP model can also model an insect employing more legs on the ground. In fact, one important insight in this study is that whenever there is more than one leg on the ground, SLIP is unlikely to work as a model. This is because SLIP assumes that the net forces on the CoM act along the single effective leg. When there are more than one leg of the ground, this constraint – forces only along the leg – severely limits the ability of a model to describe locomotion. ARSLIP removes this constraint and allows the description of forces perpendicular to the leg. Thus, the ARSLIP model is the more natural take-off point for efforts to obtain a truly general model for locomotion not only in insects but in multilegged animals in general.

The flies were reared at 25°C, and 12 hr:12 hr light:dark cycle. Ten minutes before the experiment the flies were removed from a vial and placed under CO₂ anesthesia , and their wings were detached using forceps.

We experimented with different wild-type strains to record steps at a range of walking speeds and to ensure that any general principle we discover is indeed general (at least across a range of inbred strains). These wild-type strains were w1118, Berlin K, and Oregon-R-C (or Oregon C) (Bloomington stock numbers: 5905, 8522, and 5, respectively). Table 2 shows all of the flies in our dataset and the data each fly contributed to the analyses in each figure.

Our experimental data consisted of the CoM position of the fly in all three dimensions and the position of the fly’s footholds in the horizontal plane. This section describes the acquisition and processing procedures that yield this dataset.

Tracking CoM and foothold positions

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The fly’s CoM was estimated by using the most prominent features of the fly as fiducial markers. The features were extracted on the first frame by using the minimum eigenvalue algorithm and following the extracted points throughout the video using KLT (KLT feature tracker in MATLAB [Lucas and Kanade, 1981; Shi and Tomasi, 1993; Tomasi and Kanade, 1991]). An estimated affine transformation matrix between the sets of feature points of consecutive frames was multiplied to the CoM position in the previous frame to evaluate the CoM position in the current frame (Figure 2—figure supplement 1). Next, between every pair of consecutive frames, CoM was backtracked one step. The distance between original and backtracked CoMs is a reliable measurement of the so-called forward–backward tracking error (Kalal et al., 2010). The error was small enough that we could evaluate the SLIP and ARSLIP models. The errors were also much smaller than the change in CoM position during a step (Figure 2—figure supplement 2). Therefore, the noise of the estimated CoM trajectories was small, so numerical derivations of the trajectories returned velocity trends with a small noise (Figure 2—figure supplement 2).

Foothold location was automatically detected using a series of image processing algorithms detailed in Figure 2—figure supplement 1. The basic idea was to binarize the bottom view and thin the resulting image to yield a skeleton. The end points of the resulting skeleton returned points, including the actual footholds, along with other noisy or random points. The actual footholds were robustly detected by filtering out the noisy points and extracting points that are located the furthest away from the CoM. The noise filtering was performed by removing small objects composed of fewer than 100 pixels. The legs were labeled based on the mean of each foothold trajectory in the CoM frame (see details in Figure 2—figure supplement 1).

All the processing steps are shown in Video 2.

Video 2

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We performed gait analysis using either the stance start times or the instantaneous phase.

A time series of height or speed over a tripod stance was detrended by a line that connected the values at the beginning and end of the stance phase. Finally, the maximum and minimum values of the detrended data were summed to calculate height or speed changes (Figure 4—figure supplement 1).

We chose to nondimensionalize a unit of mass by an animal’s body mass ($m$), length by an animal’s natural leg length ($R$), and acceleration by gravitational acceleration at the surface of the earth ($g$). Following this rule, we could nondimensionalize speed and spring constants as shown below:

The following system of ordinary differential equations (ODEs) (Equations 6 and 7) is derived using Euler–Lagrange equations to describe ARSLIP. A polar coordinate system was chosen for simplicity.

where $r\left(t\right)$ is the length of the single effective leg used in SLIP and ARSLIP models, $\theta \left(t\right)$ is the leg angle from the vertical axis, $k_s$ is the leg spring constant, $k_a$ is an angular spring constant, $m$ is the total mass of a fly, and $R$ is the natural leg length. Dot denotes time derivative. A detailed derivation of the ODE is presented in the supplementary section of Antoniak et al., 2019.

The best fit of ARSLIP to a given experimental trajectory was found using the Global Search algorithm (Dixon, 1978) (MATLAB Global Optimization Toolbox). The RMSEs for height and distance were individually evaluated, and the sum was used as the objective function.

$k_s$ and $k_a$ were searched within the range below (Equations 8 and 9):

These bounds were chosen empirically. $R$ and $m$ were experimentally measured for the various strains (Table 3). The length of the middle leg was measured from the still frames of the fly. Multiple measurements of a fly were averaged ($R_{real}$), and then based on the average value optimal $R$ was estimated by searching within ±10% boundary of $R_{real}$. Mass (m) was measured by averaging the mass of 10 individuals from the same genotype and gender.

Because a model with two effective legs would have too many parameters and would obscure many of the clear insights gained from modeling, we chose steps for which the duration of support by a single tripod was at least 25% of the tripod stance. This criterion does not mean that 75% of the step is spent with both tripods on the ground. Because the tripod legs are not synchronized, much of the time spent with both tripods on the substrate is the time it takes for legs from the second tripod to leave the ground. The experimentally measured initial conditions of $\dot{\theta }$ and $\dot{r}$ were used to constrain the optimal initial condition. The optimal initial condition was constrained to be within ±10% of the measurements. Because we set the foothold location of ARSLIP as the middle of the front and hind leg foothold positions, initial conditions of $\theta$ and $r$ could be determined from experimental data.

The SLIP model was fitted using a similar method under the same parameter conditions except for the absence of $k_a$ due to the lack of angular spring in SLIP. The system of ODEs for SLIP is given by

For both models, the gravitational constant $g$ had a value of $9.807m/s^2$.

The total elastic potential energy of a springy tripod (Figure 5) is given by

The variables are also enumerated in Table 1 and shown in Figure 5. In this equation, $R_{tri}$ is the natural length of the springy tripod that is optimized for each fly; $r$ is the height of the middle leg; $\theta$ is the angle that it makes with the vertical axis, which is identified with the radial and angular coordinate of ARSLIP; L is the measured spread of the tripod (Figure 7A); and k is the stiffness of a given leg. The total elastic potential energy is simply a sum of the potential energies due to each individual leg. We claim that $V_{tri}$ is equivalent to the ARSLIP potential energy, V (Equation 12), for evolution that is close to the midpoint $(R_m$), and $\theta =0$.

Because $\left|∆r/r_m\right|~0.1$ and $\left|\theta \right|~0.2$, it should be sufficient to show that the two potential energies agree with each other up to the quadratic order in fluctuations around the fixed point. This means that the first and second derivatives with respect to $r$ and $\theta$ at $r=r_m$ and $\theta =0$ are the same for both the potential energies. Specifically,

are the same for both the potential energies. The relations involving the first derivative of $\theta$ (Equation 15) and cross-double derivative involving both $\theta$ and $r$ are automatically satisfied because of symmetry (Equation 18), the latter demonstrating the independence of the radial and angular springy forces that are assumed in the ARSLIP model.

We are then left with three equations, including the first derivative of $r$ (Equation 14), and the two double derivatives w.r.t. $r$ and $\theta$ (Equations 16 and 17). The effective ARSLIP potential energy involves the parameters $R$, $k_s$, and $k_a$. Consequently, it is possible to relate ARSLIP and springy tripod using the following equations:

Thus, the parameters, $R$, $k_s$, and $k_a$ can be determined from the tripod potential energy parameters, $k$ and $R_{tri}$, and the geometric quantities $r_m$, $L$ (Equations 19–21). We obtained $r_m$ and $L$ from the geometric data for each step. We assumed that a given fly has the same $R_{tri}$ and $k,$ and fit $k_s$ and $k_a$ for all the steps of the given fly. This assumption led to a best fit value of $k$ and $R_{tri}$.

Whether the gait will be fly-like or cockroach-like is determined by the relative strengths of the radial and angular springs, which in turn depend on the tripod geometry and stiffness of the leg. In this Appendix, we will derive an expression that predicts fly-like versus cockroach-like gait based on the leg stiffness and tripod geometry.

From the ARSLIP model, we arrive at the following condition for negative horizontal acceleration at the midpoint, a characteristic of mid-stance velocity maxima:

where ${\displaystyle {\overline{r}}_m\equiv r_m/R}$, $r_m$ is the mid-stance height, and $R$ is the natural length of the effective SLIP spring. This equation is derived from a consideration of forces at mid-stance. This is derived in the Antoniak et al. manuscript.

Assuming the empirical observation that while walking the mid-stance height of the fruit flies is very close to the leg lengths that balance gravity, we have

Equation A2 simply means that the forces due to radial spring are close to the weight of the fly. Equation A2, in turn, implies that for the fly-like mid-stance maximum velocity profiles we must have

To understand what this implies in terms of tripod leg characteristics and geometry, we need to rewrite the above inequality in terms of tripod quantities. Substituting k_a from Equation 20, we have

where $\eta \equiv \frac{L}{r_m}$ and $\gamma \equiv \frac{k{R}_{tri}}{mg}$.

Thus, the above inequality (Equation A4) provides a dividing curve in the $\eta -\gamma$ plane between velocity maxima (like flies) versus velocity minima (like cockroaches).

It is also illuminating to see how geometry shapes the angular and radial spring constants. From Equations A2 and 21, we have

after some algebraic manipulations. From Equation 20, we then have

Therefore, ${\gamma }_a$ depends on $\eta$, $\gamma$, ${\stackrel{-}{r}}_{m,tri}\equiv \frac{r_m}{R_{tri}}$, and $\stackrel{-}{L}\equiv \frac{L^2}{R_{tri}}$.

The funders had no role in study design, data collection and interpretation, or the decision to submit the work for publication.

We would like to acknowledge the members of Bhandawat lab and Sanjay Sane and his lab, and Michael Dickinson for critical comments on earlier versions of the manuscript. This research was supported by NIDCD (VB), NINDS (VB), and an NSF CAREER award (VB).

© 2021, Chun et al.

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