Comparative analysis of a geometric and an adhesive righting strategy against toppling in inclined hexapedal locomotion

Terrestrial animals live in complex habitats. Consequently, many of them encounter situations where they may topple over and fall unintentionally. How they maintain their upright posture while traversing hilly habitats is therefore of particular interest in integrative biology and provides important insights with respect to the design of legged machines and physics-based computer animations.

The concept of the static stability margin SSM=min(d₁,d₂,…,d_n), which is defined as the minimum directed ‘distance of the vertical projection of the center of gravity [CoG] to the boundaries of the support pattern in the horizontal plane’ (Song, 1984), has been used to quantify the static stability for a slow walking animal with non-grasping feet (McGhee and Frank, 1968) (Fig. 1A). For a (quasi) statically stable gait, the vertical projection of the CoG is kept within the support pattern for a full gait cycle (SSM≥0) and the animal can move without toppling from one step to the next. When the vertical projection of the CoG moves out of the support base (SSM<0), a positive CoG torque emerges (M_CoG=SSM·F_CoG, where F_CoG is the body weight and has a negative value) and the animal has a tendency to topple over the critical tipping axis, which is the boundary of the support pattern nearest to the vertical projection of the CoG in the substrate plane (red line in Fig. 1A,B). A negative SSM (positive CoG torque) was observed to happen to load-carrying ants where the CoG was shifted (Merienne et al., 2020; Moll et al., 2013), and during high speeds in certain lizard species (Irschick and Jayne, 1999) and cockroaches (Full and Tu, 1991), where the animals changed their support patterns during level locomotion. Furthermore, it could also happen on steep slopes, where the distance between the vertical projection of the CoG and the support boundary diminishes (Song, 1984). Obviously, a negative SSM often does not oblige the real animals to fall while in motion. These led us to the question, how can the concept of the SSM be extended to quantify the anti-toppling behaviors of locomoting animals with grasping abilities?

A necessary condition for static stability in general is that the net torque is zero. The demand that the projection of the COG falls within the support pattern (positive SSM) is neither a sufficient nor a necessary stability criterion; the net torque can be zero if the SSM is negative or positive, provided that some feet produce a counterbalancing torque by ‘pulling’ or ‘pushing’ on the substrate. Furthermore, whereas a zero net torque can be expected for the ideal quiet stance at every point in time, it will most probably not be achieved for the legged locomotion of an animal, because it changes speed and direction, or the stepping pattern, and it moves under the influence of the gravitational pull and inertial effects. Therefore, to not tip over and fall while locomoting, the legged animal should compensate for a possible non-zero net torque with the passage of time to approach a zero integral of the net torque (angular momentum) over a stride. Such a postponed correction can be achieved, for instance, while touching down or lifting off legs, which in turn provides a new scenario of the support pattern, the forces and the torques to neutralize unintended velocity changes over time which otherwise may lead to toppling.

Thus, to describe how the animal avoids the tendency to topple over within different phases of the gait cycle, one must look at the sign and the magnitude of the CoG torque and the possibly countering leg torque about the critical tipping axis with the passage of time. For instance, the CoG torque changes permanently because of the relative movement between the CoG and the tipping axes during the animal's legged locomotion, with a possible change of sign while crossing the boundary of the support pattern. Then, if not fully balanced by the countering leg or legs, the magnitude of the CoG torque increases and accelerates the animal with some curvature along the gravitational field lines. After calculating the sum of the CoG torque and the countering leg torque for some time interval or gait phase, one can evaluate whether the animal has the tendency to tip away from the support pattern, which we term toppling (positive net torque), or towards the support pattern, which we term righting (negative net torque). Thus, while in motion, the net torque can be non-zero for a certain interval of the gait cycle (functional tipping). If the animal stops, the net torque should indeed approach zero to prevent unintended tipping.

Despite the potential to classify different locomotion behaviors and to describe functions of the gait phases based on the subsequent torque analysis, to our knowledge neither the CoG torque about the critical tipping axis on slopes nor the possibly countering leg torque has yet been quantified for climbing animals. Therefore, we propose two major strategies to not topple over in climbing animals, based on the behavior of (i) avoiding positive CoG torques by changing the geometry (keeping the SSM positive) or (ii) engaging negative leg-righting torques by grasping feet. We compared level and climbing locomotion on a 60 deg upslope in the desert ants Cataglyphis Förster 1850 and the wood ants Formica Linnaeus 1758, as the results can be related to past inclined locomotion studies in ants (Seidl and Wehner, 2008; Weihmann and Blickhan, 2009; Wöhrl et al., 2017a).

Based on the definition of SSM, positive CoG torque on slopes can be avoided through geometric changes, such as posture changes, which shift the position of the CoG. Many animals have been found to lower their body height on inclines, such as cats (Carlson-Kuhta et al., 1998; Smith et al., 1998), rats and squirrels (Schmidt and Fischer, 2011), white tree frogs (Endlein et al., 2013), stick insects (Diederich et al., 2002), as well as desert iguanas (Jayne and Irschick, 1999) and the desert ants Catalyptis fortis (Weihmann and Blickhan, 2009; Wöhrl et al., 2017a). However, this behavior was not observed in mound-dwelling iguanas (Higham and Jayne, 2004) and the wood ants Formica pratensis (Weihmann and Blickhan, 2009).

Another possibility to improve SSM – or avoid a positive CoG torque – is through stepping pattern changes, which modify the geometry of the support pattern and thus the orientation or/and the number of tipping axes. Changes in the anterior and posterior extreme positions of the legs have been observed in Colorado potato beetles on wood panels (Pelletier and Caissie, 2001) and ants on granular media (Humeau et al., 2019) during sloped locomotion. However, neither Cataglyphis fortis nor Formica pratensis was observed to implement significant inclination-dependent changes of the step length and footfall geometry with respect to the CoG on solid surfaces (Seidl and Wehner, 2008).

The destabilizing effect of the positive CoG toppling torques could be countered through strategy ii, the generation of negative leg-righting torques M_leg=r_l·F_lz′, where r_l is the effective lever arm length, or the shortest distance between the critical tipping axis and its opposing leg-surface contact point, and F_lz′ is the ground reaction force (GRF) of leg l in the normal direction to the substrate (see Fig. 1B for the front leg with l=1). This implies that at least three points are simultaneously touching the ground, that the support pattern has at least three sides, and that the feet have grasping abilities to employ adhesive forces. In the case of a triangular support pattern, only the leg which opposes the critical tipping axis can generate a leg torque about that axis, because the two remaining legs are forming the bearing for the critical tipping axis and have therefore no lever arm. Of course, the middle leg, for instance, can also employ leg torques about the line in the support plane that passes through the front leg and the hindleg and so forth. For a four-point contact scenario, the number of possible tipping axes increases accordingly, and the toppling analysis becomes even costlier. To evaluate the complete anti-toppling dynamics, one must measure the GRFs and the kinematics of each contact point over the full gait cycle. First, we could not achieve this with our current setup, because the forces of several legs could not be recorded simultaneously and independently. Second, for a three-point contact stance, this is not necessary if we are only interested in how toppling caused by the CoG can be avoided or countered.

Our proposed simplified version of this complex analysis extends the previous concept of the SSM sufficiently to distinguish between the two proposed strategies to avoid toppling: the (i) geometric strategy and the (ii) adhesive strategy. We exemplify our analysis on the simplest case, a triangular support pattern. First, to decide which leg torque should be considered for the further analysis, the identification of the critical tipping axis – which is the line where the animal would tip over, if the gravitational pull were the only force acting on the animal – must be identified. For instance, for a hexapod walking upslope along the gradient, the critical tipping axis for the three-point contact duration is most likely the axis between the hindleg and the middle leg and the countering leg would therefore be the front leg. It probably changes if the animal walks downwards or sidewards along the slope. Then, after calculating the quasi-static CoG torque and the countering leg torque for a fraction of time, one can evaluate, whether the animal has the tendency to employ the geometric strategy (i) by keeping the CoG torque negative (CoG righting), or the adhesive strategy (ii) by keeping the leg torques negative (leg righting).

During our previous study on the forces and impulses of ants (Wöhrl et al., 2017a), we found out that Cataglyphis did not employ notable pulling forces (F_lz′<0) normally to the substrate with their front legs on steep upslopes. This implies that Cataglyphis lacks the ability to generate negative leg torques to counter the possibly positive CoG torques and relies therefore on the geometric strategy. In contrast, if Formica could exert (adhesive) leg pulling forces, it is possible that Formica could generate sufficient negative leg torques to counterbalance the possibly positive CoG torques (adhesion strategy) without the need to adjust its body height or gait pattern. As the ‘normal forces that prevent the foot from lifting’ (Ramdya et al., 2017), adhesion mechanisms have been studied in a variety of animals such as geckos, frogs, beetles, stick insects, cockroaches, ants, spiders and mites (e.g. Endlein and Federle, 2015; Federle and Labonte, 2019; Gorb, 2008; Labonte and Federle, 2015), but they have not been used to calculate the leg torques with respect to the critical tipping axes. However, to evaluate the possible counterbalancing function of the legs, the leg torques must be quantified and contrasted to the CoG torques. We provide these data in our present study.

Previous observations (Seidl and Wehner, 2008; Weihmann and Blickhan, 2009) on the climbing behavior of desert and wood ants left a few open questions, which formed the basis for our present study. First, as certain animals in general, and Cataglyphis in particular, lower their body height on inclines (part of geometric strategy i), how effective is this strategy in helping it to achieve positive SSM (Fig. 1A)? A previous study discussed that Cataglyphis may hold its CoG inside the supporting polygon (Weihmann and Blickhan, 2009); however, actual quantification of the SSM is missing. Second, if Formica did not employ either of the geometric strategies (i), as suggested by the previous observations, does it experience positive CoG torques (negative SSM) on steep slopes, and if that is the case, does it counter the possibly positive CoG torques by employing negative leg torques (adhesive strategy ii)? To answer these questions, we analyzed the SSM, the height of the CoG, the effective lever arms, the area of the support pattern, the CoG and leg torques on level locomotion (control group) and on a 60 deg upslope for Cataglyphis and Formica. Finally, we examined and compared the tarsal structures of the legs to relate their morphological structures to their differing climbing strategies.

Individual worker ants of the genus Formica, belonging to the Formica Linnaeus 1761 species group, were randomly collected from the outside area of their nest close to the village Golmsdorf (Germany) in autumn 2012. In total about 60 individuals were used for the measurements on level locomotion and the 60 deg upslope and sheltered in a formicarium with soil and materials from their natural habitat. The ants were released back into the field after the measurements. A colony of ants of the genus Cataglyphis, belonging to the species Cataglyphis fortis (Forel 1902) (identified using Wehner, 1983), from Menzel Chakar (Tunisia) with more than 600 ants, was borrowed from C. Bühlmann (Max Planck Institute for Chemical Ecology, Jena) from October 2012 to March 2013. Both ant species were fed with honey, water and insects ad libitum. One half of each formicarium was illuminated with a 60 W daylight lamp at around 28°C from 07:00 h to 19:00 h.

The two ant species in our measurements are comparable in body mass, size and temporal measures. The studied Cataglyphis ants (N=37) had a body mass of m=24.5±3.5 mg (mean±s.d.) and a thorax length (altitrunk including petiole) of l=3.9±0.3 mm, as compared with Formica ants (N=29), m=20.2±2.5 mg and l=3.6±0.2 mm. Of the analyzed Cataglyphis and Formica ants in our experiment, 97% walked by definition, as their mean duty factor of the front, middle and hindlegs was above 0.5.

We used the same experimental setup and procedure for both ant species and conducted the experiments in our indoor laboratory between 2012 and 2013. Related data of the locomotion experiments and further details of the setup can be found in previous publications (Reinhardt and Blickhan, 2014a,b; Wöhrl et al., 2017a,b).

To carry out the experiments, an ant was randomly taken with a plastic tube and placed at the beginning of a 90 mm (length), 25 mm (width) and 30 mm (height) confined running track covered with ordinary millimeter paper. A three-dimensional 4×4 mm force platform was custom built in the center of the track and covered with the same millimeter paper. It used PVC/semiconductor strain gauges (KSP-3-120-F2-11, Kyowa, Tokyo, Japan) and resolved forces in the running direction F_x=5.4 μN, lateral direction F_y=2.9 μN and normal direction F_z=10.8 μN with natural frequencies of f_x=380 Hz, f_y=279 Hz and f_z=201 Hz (Reinhardt and Blickhan, 2014b). A data acquisition system (MGCplus, 1200 Hz, Hottinger Baldwin Messtechnik, Darmstadt, Germany) sampled the signals received by the force platform and amplified them.

One top-view Photron Fastcam SA3 (San Diego, CA, USA) captured the animals at 500 Hz dorsally and laterally through the right-angle glass prisms which were assembled at one side of the tunnel and functioned as mirrors. The kinematic data were obtained semi-automatically with DigitizingTools 20160711 (Hedrick, 2008) and MATLAB R2015b (The MathWorks, Natick, MA, USA). The body mass of each ant was measured with a precision scale (±0.1 mg, ABS 80-4, Kern & Sohn) immediately after it stepped on the force platform. The measured ants were kept separated from the ants in the formicaries and returned to the formicaries at the end of every daily series of measurements.

In total, 753 individual leg force measurements of the two ant species on level locomotion and 60 deg upslope were recorded. In the first round of data screening, we excluded trials in which the ants stopped on the force plate or touched it with their gasters. As we could only measure the force of one leg from each run, the tipping behavior of the gait phase where more than three legs were touching the ground could not be analyzed. Therefore, we focused our analysis on the three-feet-only contact durations, which is not the same as the tripod duration (Fig. 1C), and excluded trials where no three-feet (only) contact stance was captured. As the ants were returned to the formicaries at the end of every measurement day, there was the chance of randomized pseudoreplications. To minimize bias by these pseudoreplications, we further reduced the dataset and chose only the first measurement per ant per leg per day for the force, torque and impulse data (detailed sample size statistics are given in Table 1) and maximum one measurement per ant per slope per day for the SSM data (sample size statistics are given in Table 2). However, a preceding study relativized the effects of possible pseudoreplications in force and kinematic measurements in Formica (Reinhardt and Blickhan, 2014a).

The time series data of the measured ground reaction forces and calculated torques were time normalized with linear interpolation to the durations of the three-feet contacts. The mean three-feet contact duration on the 60 deg upslope was 15±9 ms for Cataglyphis and 24±7 ms for Formica.

As the slope-dependent changes of the gaster angle are relatively small for both ant species (Weihmann and Blickhan, 2009), the CoG was calculated according to previous studies at a distance of 20% of the thorax length cranially from the thorax–petiole joint for Cataglyphis (McMeeking et al., 2012) and 0.3 mm cranially from the same joint for Formica (Reinhardt and Blickhan, 2014a). The CoG and the support pattern formed by the three-feet contact were vertically projected into the horizontal plane to obtain the SSM (Figs 1 and 2).

MATLAB 9.6.0 R2019a and R 4.0.2 (http://www.R-project.org/) were used for data analysis. Estimation graphics (Ho et al., 2019) with median differences (Δ), 5000 bootstrap resampled distributions and bias-corrected 95% confidence intervals (CI) (R package dabestr v0.3.0) were used to quantify the total effect of slope or species.

One specimen of each species, Cataglyphis fortis and Formica rufa, was fixed in 70% ethanol. The legs were removed with fine forceps, dehydrated in a rising ethanol series (80%, 90%, 96%, 100%) and transferred to acetone for 1 h, with the acetone exchanged every 20 min. The samples were dried at the critical point in liquid CO₂ subsequently with an Emitech K 850 Critical Point Dryer (Sample Preparation Division, Quorum Technologies Ltd, Ashford, UK). The dried samples were glued on needles and sputter coated with gold in an Emitech K 500 (Sample Preparation Division, Quorum Technologies Ltd) and attached to a rotatable specimen holder (Pohl, 2010).

The scanning electron microscope (SEM) micrographs were taken with a Philips ESEM XL30 (Philips, Amsterdam, The Netherlands) equipped with Scandium FIVE software (Olympus, Münster, Germany). The SEM micrographs were assembled as an image plate using Adobe Photoshop^® CS6 (Adobe System Incorporated, San Jose, CA, USA) and labeled in Adobe Illustrator^® CS6 (Adobe Systems Incorporated).

To identify whether the ants employed the geometric strategy on steep upslopes, we first plotted the vertical projection of the CoG in relation to the support pattern and deduced the location of the critical tipping axis (Fig. 2A). For Cataglyphis, the vertical projections of the CoG fluctuated around the vertical projection of the critical tipping axis at the point in time where the SSM reached its minimum (Fig. 2A, zenithal view). The critical tipping axis is the red line in Fig. 2A (normal view) formed by the middle leg and the hindleg for all upslope measurements and both ant species. For Formica, the vertical projections of the CoG were outside of the support pattern for the entire three-point contact duration.

To validate the first hypothesis, we first took the level locomotion geometry measurements of the ants and calculated their would-be SSM values for a hypothetical 60 deg upslope by rotating the level locomotion geometry around the y-axis by 60 deg (Fig. 2B, control group). Then we compared the would-be median SSM values with the actual median SSM values calculated from the geometric measurements on a 60 deg upslope. We expected that by changing its geometry, Cataglyphis could improve its SSM from a negative would-be SSM to a positive actual SSM. Vice versa, if Formica did not change its geometry, the actual SSM on the upslope should not differ from the would-be SSM estimated from the level locomotion measurements.

For Cataglyphis, the measured total effect of slope on the SSM was an improvement of +1.9 mm with a 95% CI that ranged from 0.8 to 2.5 mm. This can be interpreted as a significant change at an alpha level of 0.05 as the 95% CI does not overlap with zero (Fig. 2B). For the Formica sample, there was no significant difference at an alpha level of 0.05 between the level locomotion and the upslope measurements.

The change of the SSM of Cataglyphis was effected by a lowered CoG of −1.7 mm [95% CI −1.9, −0.7] and by an increased area of the support pattern of +17 mm² [95% CI 9, 29] (Fig. 3). The effective lever arm, as a link between the geometric and the adhesive strategy, did not change for both species significantly on an alpha level of 0.05. For Formica, none of the analyzed geometric parameters in Fig. 3 changed, implying that a potential stepping pattern change did not offset a possible change in the body height.

To test the second hypothesis, we calculated the quasi-static CoG torques from the product of SSM and body weight M_CoG=SSM·F_CoG, as well as the quasi-static torques of the front legs M_leg. The leg torques were calculated from the product of the effective front leg lever arm r_l and the front leg GRF F_lz′ (Fig. 4A) in the normal direction M_leg=r_l·F_lz′. Thereby, Formica should exert negative front leg torques over a longer period and/or with higher magnitudes if our hypothesis holds. Furthermore, the sum of the two torques M=M_CoG+M_leg, as a measure of their tendency to tip outwards (toppling) or inwards (righting) with respect to the supporting triangle, should not be greater than zero if the leg torques are sufficient to counter the possibly positive CoG torques.

The single leg force measurements indicated that Formica pulled stronger and longer with their front legs on the substrate (F_lz′<0) than Cataglyphis (Fig. 4A). Cataglyphis could hardly employ any negative leg torques over the whole three-feet contact duration (blue line in Fig. 4B) because of the negligible pulling forces of the front legs. In contrast, Formica pulled normally to the substrate with its front legs throughout the entire three-feet contact duration (Fig. 4A) and generated negative leg torques (leg righting) to counter the positive CoG torques (Fig. 4B).

For Cataglyphis, both M_CoG and the sum M_CoG+M_leg were negative throughout the three-feet contact duration (geometric strategy), whereas M_leg was close to zero (Fig. 4B). This implies that Cataglyphis employed its geometric strategy successfully to avoid tipping backwards (toppling). Formica experienced the risk of positive M_CoG for the entire three-feet contact duration, but the overall static torque M was also kept below zero because of the presence of relatively high and continuous negative leg torque M_leg (Fig. 4B). This implies that Formica obviously tolerated the risk of toppling backwards by countering the positive CoG torques by negative leg torques (adhesive strategy).

Our scanning electron microscope micrographs of the front legs (Fig. 5) indicated that arolium, claws, spines, and long and short setae are present on the tarsomeres of both ants. The main differences are the size of the arolium, the width of the tarsomeres, as well as the number, distribution and length of the setae. The Formica sample has much wider tarsomeres and many fine short setae, while the Cataglyphis sample has a substantial reduction of the arolium and distinctly fewer short setae. A less conspicuous difference is the presence of tiny folds in the surface of the arolium in Formica, whereas the arolium is completely smooth in Cataglyphis (Fig. 5, insets).

In order to compare and explain distinct hexapedal locomotion behaviors for traversing steep slopes, we quantified SSM as well as CoG torques and leg torques over the three-feet contact duration for Cataglyphis and for Formica on a solid 60 deg upslope covered with millimeter paper. The results revealed that Cataglyphis primarily engaged in geometric changes (strategy i) to improve its SSM (Fig. 2B). Subsequently, it did not experience any positive CoG torques (toppling backwards) during the three-feet contact (Fig. 4B). The normal pulling and pushing forces of the front legs were negligible (Fig. 4A) and did not contribute notably to the righting (negative torques) nor did it enforce the overall toppling risk (positive torques) (Fig. 4B).

In contrast, Formica’s front legs were able to pull relatively strongly (Fig. 4A), which contributed to stronger negative leg torques (Fig. 4B). These were sufficient to counterbalance the positive CoG torques (Fig. 4B). In other words, Formica tolerated a higher risk of toppling backwards on upslopes. Instead of changing the geometry to improve its SSM, it chose to pull more strongly on the substrate with its front legs opposite the critical tipping axis (adhesion strategy ii).

As two common results, the sum of the quasi-static torques M_CoG+M_leg, remained negative for almost the entire three-feet contact for both ants (Fig. 4B), and the critical tipping axis was in all measurements the line that travels through the middle leg and hindleg tarsi. These imply that the ants experienced tipping torques towards the supporting area during their three-feet stance (righting). To not collapse under the observed negative net torques – which accelerated the ants towards the substrate above the critical tipping axis – the remaining three feet of the current swing phase should counterbalance the ants’ posture later on by their following touch down with a new scenario of the support pattern, the forces and the torques. Furthermore, an impactful touch down could have advantages for the required load (positive normal force) on their tarsi to improve traction (Labonte and Federle, 2013).

Despite this, we observed qualitatively that Formica traversed steeper and even vertical inclines seemingly effortlessly, whereas Cataglyphis showed difficulties in traversing the 60 deg upslope. This is most probably attributed to the physical limitation of the geometric strategy, because the SSM cannot be maintained at a positive value once the slope crosses a certain steepness related to the geometry of the ant. In contrast, the adhesive locomotion enables climbing on a larger range of slopes, including vertical or inverted substrates (e.g. Federle and Endlein, 2004; Gorb et al., 2007), which is impossible on the basis of the geometric strategy.

Another advantage of the adhesion strategy is that it decreases the risk of slipping on steep slopes. As the slope becomes steeper, the ratio between the normal forces and the associated downhill or tangential forces decreases for quasi-static model assumptions. As the frictional forces are proportional to the normal forces according to Amontons’ laws of friction, the frictional forces also become smaller and the risk for slipping becomes greater. To reduce the risk of slipping, the animal can increase the frictional force by pressing against the substrate with the hindlegs (increasing the normal pushing force) while pulling more strongly with its front legs. In contrast, with a lowered CoG of the geometric strategy (i), the tangential forces on the legs could be even further increased, which could easily surpass the maximum possible frictional force per leg and cause slipping. Formica does not reduce its CoG height, such that the time integral over the normal forces as a measure of the effective force applied during the three-point contact, or the impulse surface angle, remained relatively high compared with the time integral of the tangential forces (Fig. 6A). Furthermore, by exerting stronger pulling forces with its front legs, Formica might also push more strongly in the normal direction with at least one of the remaining legs to partly counter this impact. In particular, the stronger normal forces over time exerted by Formica can result in more bending of its many extremely fine setae (Endlein and Federle, 2015) (Fig. 6B). Subsequently, the bending of the setae can lead to more microscopic contact points (Gorb et al., 2007) and stronger nanoscopic attractive normal forces such as electrostatic forces (Izadi et al., 2014), van der Waals forces (Autumn et al., 2000) or capillary forces (Gorb et al., 2007). The increased number of contact points should hence result in higher maximal friction forces and a better performance in accelerating or traction. This was also observed in another climbing ant, Oecophylla, which pressed its dense tarsal hair against the substrate to intensify the friction forces (Endlein and Federle, 2015). Therefore, the upright body posture, the intensified pushing onto the substrate with the hindlegs (Fig. 4A) plus the presence of more fine tarsal hairs in Formica (Figs 5 and 6B) should explain its higher gradeability (the highest inclination an animal can ascend) and faster mean climbing speed (93±30 mm s⁻¹ compared with 60±22 mm s⁻¹ for Cataglyphis).

Despite the advantages of the adhesion strategy with respect to the gradeability, Cataglyphis did not exert strong pulling forces with its front legs. Our microscopic examination revealed that its front legs (Fig. 5), like the hindlegs (Fig. 6B), had a smaller arolium, slimmer tarsomere and fewer fine short setae compared with those of Formica. In addition, the surface of the arolium was completely smooth in Cataglyphis but Formica had fine micro-folds in the arolium. Differences in the tarsal structures are well documented as varying factors between ground-dwelling and climbing species (Billen et al., 2017; Orivel et al., 2001). For example, the ground-dwelling ants Brachyponera senaarensis lack fine hair on the ventral tarsal surface and the arolium gland is very small compared with that of the climbing arboreal ant Daceton armigerum (Billen et al., 2017). Thus, unlike Formica, the ground-dwelling Cataglyphis might not have the ability to exploit sufficiently the adherence capabilities of their tarsal and pre-tarsal structures on the millimeter paper and is therefore forced to avoid the toppling risk by lowering its CoG. Moreover, in salt pan habitats with their dry, hot and granular ground, a large arolium may not benefit Cataglyphis as much as Formica in its surroundings, with probably more stimuli to climb up vertically or inverted on plant components with solid, smoother and colder surfaces. A potential wet adhesive functionality of the Cataglyphis arolium could also lead to faster dehydration in desert environments and to dust adherence to their tarsal structures. In addition, a larger arolium gland or the potentially attached dust particles may hamper the required fast swing-leg retraction time for their fast running. In contrast, the claws may still help Cataglyphis to generate traction or climb in their nest on the granular material.

Although fine dense hair on the tarsal euplantulae (‘heel’) and a prominent pre-tarsal arolium (‘toe’) are advantageous for climbing hexapoda (Bullock et al., 2008; Clemente and Federle, 2008; Labonte and Federle, 2013), the associated energetic cost of locomotion may be disadvantageous for ground-dwelling species. Furthermore, the longer spines on the tarsi of Cataglyphis could possibly function to store and release elastic energy for its well-known fast running capability (Wehner, 1983) better than shorter and finer setae. Apart from these differences in their tarsal structures, Cataglyphis may not yet be accustomed to utilizing their neuromuscular–adhesive system fully to switch from a ‘pushing’ gait to an adhesive ‘pulling’ gait on our slopes, which were covered with millimeter paper.

Based on the approach to calculate torque data with respect to the critical tipping axis, we were able to classify two different locomotion behaviors – a geometric strategy of the desert ants and an adhesive strategy of the wood ants – to avoid toppling when climbing up steep slopes. This helped us to evaluate the effectiveness and the limitations of the body height adjustment by Cataglyphis to maintain its upright posture in inclined locomotion and to identify the righting behavior of Formica on steep slopes. The results showed that Cataglyphis desert ants turned potentially positive CoG torques (toppling) to negative ones (righting) with their geometric strategy but exhibited difficulties in traversing slopes steeper than 60 deg with this strategy. Formica wood ants tended to ignore the increasing influence of the positive CoG torques on the steep upslope and instead engaged an adhesive leg-righting strategy by pulling with their front legs on the substrate. The latter strategy is more advantageous for climbing with respect to the gradeability. Furthermore, its associated relatively steep leg impulse substrate angles do not increase the risk of slipping as much as relatively flat leg impulse substrate angles do. This strategy is thus able to exploit the capabilities of the sophisticated adhesive system of the tarsal structures by maintaining a relatively high normal force on the setae. Therefore, Formica benefits probably more from employing only one strategy instead of two, because the maintenance of its geometry keeps the normal load relatively high on its hindleg against slipping compared with a change of its geometry to reduce the risk of toppling.

Our analysis exemplifies how the concept of the SSM could be extended for inclined locomotion with grasping feet to distinguish between a geometric and an adhesive locomotion behavior to prevent toppling. This biomechanical analysis may be linked to several adjacent physiological topics. For instance, a problem of potential general interest may link metabolic rate with the locomotion of animals in their habitats in terms of which strategy is energetically costlier, or in the end more relevant for survival in their ecological niche. A potential conflict between adhesion and locomotion would favor the geometric strategy energetically. In contrast, the lowered posture with its larger supporting area could also increase metabolic rate as a result of the varied operation of the muscles, or the possibly costlier adhesion strategy could increase the gradeability and subsequently enable steeper environments to be inhabited. A comparative analysis, based on the methods of a study on climbing Camponotus ants, for instance (Lipp et al., 2005), could relate the CO₂ release to the geometric and adhesive climbing behavior to approach this question.

Last but not least, our analysis relies on two main simplifications, which could be addressed in further studies with an advanced setup to gain elaborated insight into the tipping dynamics and behavior of the full gait cycle. First, we narrowed our analysis to the three-feet stance duration, because we could not measure the force of several legs simultaneously and independently. To expand and verify the results further to the complete gait cycle, one must record all the GRFs for at least one full cycle. Thereby, the time integral of the net torques (net angular momentum) of the transitions between the three-feet stances should counter the remaining negative net angular momentum from the three-feet stance (Fig. 4B). Second, our analysis ignored the existence of inertial effects, because the calculation of the inertial torques relies on a higher spatial resolution of the tracked points compared with what we could have provided with our data to present reliable results on the inertial torques. Theoretically, the inertial effects could also help the animals either to right their posture (negative inertial torques) or to enforce the risk of toppling over (positive inertial torques) during sudden changes of their speed or direction (Raibert et al., 1995). However, our results imply that no negative inertial torques were required to not topple over the critical tipping axis, and possibly positive inertial torques did not constrain the animals to a backward movement.

We would like to thank C. Bühlmann for lending us a Cataglyphis fortis colony from October 2012 to March 2013. We thank the two reviewers, the editor, D. Labonte and S. Cai whose critical reading and suggestions helped improve and clarify the manuscript.