Mechanics of generating friction during locomotion on rough and smooth arboreal trackways

Most small quadrupedal mammals are at least somewhat proficient at moving on or under tree trunks, branches, twigs and shrubs(Jenkins, 1974). Such substrates are commonly encountered by any animal that lives where vegetation grows. The texture of such arboreal substrates varies considerably. Some have rough bark, and animals moving on them can easily generate enough friction between the substrate and their hand or foot. The animal can also (or instead)use claws to generate or take advantage of surfaces that are relatively perpendicular to the direction of locomotion(Cartmill, 1985). On smooth arboreal surfaces, creating and maintaining a stable contact with the branch is probably more challenging. There may be fewer places for claws to engage with the surface, and generating adequate friction force may be difficult because the coefficient of static friction (μ_s, which is a dimensionless number expressing shear force divided by normal force) is lower.

When generating friction force (μ_s multiplied by normal force) is the method (or an important method) of maintaining contact with a substrate, there are three ways for the animal to generate enough friction force. First, it can increase the normal force, which is directed perpendicular to the substrate. On an arboreal substrate, which is roughly cylindrical, a normal force runs from the center of pressure of the hand or foot toward the centerline of the long axis of the branch(Fig. 1). Normal force can be increased by muscular effort; for example, contralateral limbs can adduct,squeezing the branch between them and generating greater normal force. If the contact is near the top of the branch (that is, near the dorsal-most part of a long, narrow, horizontal cylinder), then considerable normal force will be generated from vertical force. The vertical component of SRF is typically the largest component, and when the limbs contact the top of the branch, most or all of the vertical force is normal to the branch surface. The second way to generate adequate friction force is to increase μ_s. It might be possible for an animal to increase the effective μ_s by changing the shape or moisture content of its hand or foot(Haffner, 1998). Finally, the animal can decrease the need to generate friction force by decreasing the shear forces. Vertical and mediolateral components of the substrate reaction force (SRF) typically have a shear component whose magnitude corresponds with the location of the center of pressure on the branch cylinder(Fig. 1). In addition, the entire craniocaudal (fore–aft or anteroposterior) force component is a shear force. Finally, the torque around the long axis of the branch, which is generated separately from SRFs (abbreviated as τ_CC,musc,which stands for craniocaudal torque generated by muscular effort)(Lammers and Gauntner, 2008),also generates shear force (applied around a moment arm). Shear force can be decreased by moving more slowly (which should decrease braking and propulsive craniocaudal forces), by placing the limb on the top of the branch (which decreases the shear component of the vertical force) or by decreasingτ _CC,musc. It is possible that an animal might use all three methods (increasing normal force, increasing μ_s and decreasing shear force) to ensure adequate friction force with a particular substrate.

In this paper, I explore the first and third methods of friction force generation. Two questions are addressed. First, how does the texture of an arboreal trackway affect the means by which a small, generalized quadrupedal mammal produces friction force between the substrate and the manus or pes? To answer this question, I compared normal and shear forces between trackways and between the hand and foot. In this way, I could determine whether the opossums tended to reduce shear forces, increase normal force or some combination of these strategies. Second, how does contact location of the hand or foot affect friction force generation? A limb contact on the top of the branch should generate relatively large amounts of normal force because the large vertical force, supporting body weight, contributes more to normal force as the center of pressure moves to the top (dorsal surface) of the cylindrical arboreal trackway. Likewise, when a limb contacts the sides (or near the sides) of a branch trackway, the shear force (as measured by peak and median shear rate)should increase because the large vertical force is mostly tangential to the branch's surface. To answer this question, I examined the covariance between contact location and the required coefficient of friction (μ_req,which is the ratio of shear force to normal force), shear force and normal force.

Five gray short-tailed opossums [Monodelphis domestica(Wagner, 1842)] were used for these experiments. All animal care and procedures conform to the Cleveland State University Institutional Animal Care and Usage Committee guidelines and NIH guidelines.

The opossums were encouraged to run across two cylindrical trackways constructed of wooden dowel rods, 2 cm in diameter and about 2 m long. The first (rough) trackway was covered with 60-grit sandpaper, which provided a large μ_s between the trackway and the hand or foot. The second(smooth) trackway was covered with thick paper (actually, sandpaper with the grit side facing inward). Thus, the trackways were identical except for the surface texture. It is unlikely that the animals could sink their claws into the substrates because the glue used to attach paper to the cylinders was very hard. The animals ran into an enclosure placed at the end of the trackway.

A 3.8 cm-long region of each trackway was instrumented to measure the vertical, craniocaudal (fore–aft or anteroposterior) and mediolateral components of the SRF (F_V, F_CC and F_ML, respectively). The force pole design is described more fully by Lammers and Gauntner (Lammers and Gauntner, 2008). Force data (in the form of voltage changes) were collected at 2000 Hz with a National Instruments signal conditioning block(SC-2345 with SCC-SG04 and SCC-DO01 modules) and a LabView 7.1 virtual instrument (National Instruments, Austin, TX, USA). Each force trace consisted of a right forelimb contact followed by a right hindlimb contact; typically,there was little or no overlap between fore- and hindlimb. Voltages were then filtered using a moving average filter (average of points 1-33, then points 2-34, etc.) in Microsoft Excel (Redmont, WA, USA). This filter removed 60 Hz noise, but it had a negligible effect on the timing and magnitude of peaks and valleys in the voltage record. A second LabView virtual instrument was used to convert voltages into force [measured in Newtons (N)]. For the final analysis,the body weights of individual opossums were taken into account by converting Newtons into body weight units (BW units).

Because τ_CC,musc has a distance factor in addition to force, F_shear is also a torque around the long axis of the branch. The distance from the manus/pes center of pressure to the long axis of the branch was 1 cm, and torque was measured in BW units cm^–1. Thus, the numbers are not affected by the distance component of torque, and I treated F_shear the same as F_normal. Slipping from the substrate is most likely to occur during the peak F_shear; therefore, peak F_shear (and F_normal) were calculated. Theμ _req, which is the ratio of F_shear to F_normal, was calculated as a measure of how likely the animal was to slip from the substrate. A higher value of μ_reqimplies a higher F_shear and/or a lower F_normal. The animal is most likely to slip whenμ _req is at its peak; therefore, I calculated the peakμ _req. Finally, to gain an overall measure of shear and normal forces and μ_req during each step, I calculated the median F_shear, F_normal andμ _req.

I measured speed with a high-speed video camera (120 Hz, JVC DVL 9800). I digitized the tip of the nose each time a forelimb (right or left) touched down and then calculated the craniocaudal displacement that occurred during each step. Craniocaudal displacement was divided by time to acquire step speed, and step speeds were averaged to calculate the speed for the individual trial. If step speeds were always within 15% of trial speed, then I accepted the trial as steady speed. This process eliminated many trials, resulting in small sample sizes for some individuals(Table 1).

Systat version 11 (Richmond, CA, USA) was used for all analyses. Speed and limb phase were compared between substrates using a two-sample t-test. To determine if speed was a significant predictor of shear and normal forces and the μ_req, I calculated Pearson correlation coefficients for each limb on each substrate. Because speed was generally poorly correlated with peak and median forces and μ_req, speed was ignored in subsequent analyses. To determine the effects of substrate texture and limb pair (hand versus foot) on how friction force is generated, I used three analyses to compare the peak and median F_shear and F_normal, peak and medianμ _req, and θ between limbs and between substrate textures. Duty factor was also compared between limbs and between substrates in the same way. First, for each measurement (e.g. peak shear force), the measurements were averaged within individual opossum. Thus, each individual was weighted equally, and a fixed-factor two-way analysis of variance (ANOVA) was used to determine significant differences between substrates and between limb pairs.(Because samples were very small for some individuals, a repeated-measures analysis could not be performed; Table 1.) In the second analysis, I calculated the percentage contribution of fore- and hindlimbs to each force and μ_reqmeasurement [forelimb/(forelimb+hindlimb)× 100%]. These percentages were compared between substrate textures using a two-sample t-test. In the third analysis, I used two-way analysis of covariance (ANCOVA), with the polar coordinate of the manus and pes contact locations in the transverse axis of the branch trackways (θ) as the covariate, to determine how forces andμ _req were affected by limb contact position among the substrate and limb groups. When slopes were homogeneous between groups, the least-squares means were compared. When slopes were not homogeneous, the slopes were calculated using least-squares regression.

Speed was significantly greater on the rough arboreal trackway (43±3 cm s^–1; mean ± s.e.m.) compared with the smooth trackway (16±2 cm s^–1; P<0.0001). Limb phase was also significantly greater on the rough trackway compared with the smooth trackway (45±1% versus 30±1%; P<0.0001) (Fig. 2). Duty factor was higher in forelimbs than hindlimbs (P=0.0228) and higher on the smooth trackway regardless of limb (P=0.0002;interaction term, P=0.43) (Table 2). Speed was usually poorly correlated with peak and median shear and normal forces, and peak and median μ_req. Pearson correlation coefficients were low; the maximum coefficient was 0.61, and minimum was–0.59; least-squares regression yielded only six significant regressions out of 24 possible relationships. Although relationships between speed and other variables were weak or non-existent, usually the coefficients were negative when comparing F_shear and μ_req with speed, and positive when comparing F_normal with speed.

Typical SRFs, μ_req, F_shear and F_normal are shown in Fig. 3. ANOVA revealed no significant differences in F_shear between limbs or between substrates(P≥0.0910) (Table 1; Fig. 4). Peak and median F_normal was significantly higher in forelimbs compared with hindlimbs (P≤0.0001). Peak and median F_normal were not significantly different between trackway surface textures (P≥0.2769). Peak μ_req was higher in hindlimbs, but there were no significant differences in μ_reqbetween rough and smooth arboreal trackways (P≥0.3162). Meanθ was significantly higher in forelimbs than hindlimbs(P=0.0393), but there was no significant difference in θbetween substrates (P=0.5879). There were no significant differences between substrate textures with respect to the percentage contribution of the forelimb to the total fore- and hindlimb contribution of force orμ _req (P≥0.1589)(Table 3).

Two-way ANCOVA was used to determine if peak and median F_shear and F_normal, and peak and medianμ _req, changed with limb contact position (θ) and if this variation differed between substrates and/or between fore- and hindlimbs(Table 4; Fig. 5). The first part of this test determined if the slopes (e.g. peak F_shearversus θ) were homogeneous among groups. In the case of peak and median F_normal, the slopes were significantly different between limb pairs (peak F_normal slopes were 0.0071 for forelimbs and 0.0034 for hindlimbs; median F_normal slopes were 0.0060 for forelimbs and 0.0025 for hindlimbs; P≤0.0024). The second part of the ANCOVA determined if slopes were significantly different from zero and if least-squares means were significantly different. Median F_shear was not correlated with θ. When θ was taken into account by ANCOVA, patterns were essentially identical to those revealed by the two ANOVA comparisons. The only difference was that peak and median F_normal were not significantly different between rough and smooth trackways (P≥0.0651).

During locomotion on any surface, the effective friction force must be equal to or greater than the sum of all shear forces. Effective friction force can be increased by escalating the normal force and by increasing theμ _s. Gray short-tailed opossums are considered terrestrial(Cartmill, 1972; Lee and Cockburn, 1985), and they are not particularly proficient at traveling on arboreal substrates [as measured by the significant decreases in speed, increases in duty factor, and changes in footfall sequence when switching from terrestrial to arboreal substrates (Lammers and Biknevicius,2004)]. Nevertheless, it seems that they have features that enable them to negotiate arboreal substrates of various textures(Cartmill, 1974; Hamrick, 2001). Thus, Jenkins'supposition that most small mammals must be at least somewhat well-adapted to traveling on arboreal substrates (Jenkins,1974) is supported.

Comparative data describing the effects of substrate texture on locomotion are few, and the kinetic adjustments that cows, dogs, geckos, humans and opossums use to remain stable on relatively slippery surfaces are probably rather different because of body size, bipedalism versusquadrupedalism, and having very different autopodial surfaces. A study by Phillips and Morris (Phillips and Morris,2001) found that dairy cows walking on surfaces with lowμ _s (0.33) walked slowly and with relatively upright limbs(higher stride frequency, but short step length). On medium μ_sfloors (μ_s=0.42 and 0.49), they walked fastest and with the greatest step length. On high μ_s floors (0.74), they slowed down again, presumably to limit the amount of wear on their hooves. The relatively upright posture of the limbs should increase normal force and decrease shear force because body mass is being accelerated in a horizontal direction (or more force is redirected through the limbs in a horizontal direction). Thus,this posture (and decrease in speed) should also decrease shear forcewhen the cows move on relatively slippery surfaces. By contrast, Kapatkin et al. found no differences in peak forces or impulses (vertical and craniocaudal components of the SRF) generated by dogs trotting on linoleum versuscarpet-surfaced trackways (Kapatkin et al.,2007). Therefore, dogs make no change in shear force, at least when running on carpet versus linoleum. Finally, Tokay geckos may increase shear force (up to twice the adhesive force, depending on the direction of shear force relative to the setae) within arrays of thousands of setae to cause powerful friction and adhesion forces that can prevent a gecko from slipping from smooth, dry glass(Autumn et al., 2006). My comparison among these three groups is simplistic, mainly because the data collected and the taxa are so varied. Nevertheless, it demonstrates astonishing diversity among tetrapods in how they cope with traveling on relatively smooth surfaces.

When the μ_req that I calculated is compared with values ofμ _req and μ_s from the literature, my values ofμ _req appear surprisingly high. Cartmill reported aμ _s of ∼0.3–0.4 for leather on clean wood(Cartmill, 1974). Redfern et al. reported μ_req data for humans walking down ramps; even at the steepest ramp angle (20 deg.), μ_req was less than 0.6(Redfern et al., 2001). Theμ _req represents F_shear/F_normal; F_shearwas rather high in my experiments because the vertical force had a considerable shear component when the hands and feet contacted the sides of the branch. The values of θ might have also been underestimated (that is, they show center of pressure of the hand and foot as closer to the sides of the branch than they actually are); this underestimation would occur if the center of pressure was closer to the medial side of the hand and foot as opposed to the center. Furthermore, the τ_CC,musccontributed substantially to the F_shear. Finally, it seems likely that not all F_normal could be measured. When the opossums gripped the branch, the hand and foot most likely generate some internal squeezing force, some or all of which is normal to the cylindrical surface. My equipment could not measure this source of F_normal. It is possible that μ_req is higher in the hindlimbs because the opposable hallux of the foot allowed greater F_normal to be generated (but not measured). It seems that designing a force pole that can measure center of pressure in the transverse plane and the squeezing forces will be quite valuable.

Peak and median F_normal were higher in forelimbs compared with hindlimbs. This result is most likely caused by two important factors. First, vertical force is greater in the forelimbs than in the hindlimbs(measured by calculating peak vertical force and the vertical impulse)(Lammers and Biknevicius,2004). This pattern is the result of the location of the body center of mass, which is closer to the forelimbs than to the hindlimbs(Lammers et al., 2006). Friction force is the result of F_normal ×μ _s; thus, the opossum's weight is acting to increase the friction force generated by both limbs, but more so in the forelimbs. The second factor is where the fore- and hindlimbs contact the cylindrical trackway around its circumference. Regardless of substrate texture, forelimbs typically contacted the branch trackways at about 63 deg., whereas hindlimbs contacted at about 42 deg. Because forelimbs contacted the branch at a point closer to the top of the branch, the very large vertical forces should contribute more to F_normal in the forelimbs than in the hindlimbs.

Peak and median F_shear were not significantly different between limb pairs. At least three important factors contributed to this pattern (Fig. 6). The first major factor is the interaction between vertical force and the contact location. Because the hindlimb contacts the branch more laterally, a greater proportion of the vertical force generated by the animal's body weight is tangential to the branch surface (that is, shear force). But the forelimbs support more body weight (Lammers et al.,2006), and so vertical force is greater. Thus, the hindlimbs have a smaller vertical force contributing proportionally more to shear force,whereas the forelimbs have a larger vertical force contributing proportionally less to shear. The second major factor contributing to the lack of differences between fore- and hindlimbs with respect to shear force is the interaction between the shear component of mediolateral force and theτ _CC,musc. Lammers and Biknevicius found that fore- and hindlimbs exerted net mediolateral impulse in a medial direction(Lammers and Biknevicius,2004). Thus, contralateral limbs squeeze the branch between them. At the same time, the τ_CC,musc was found to be in the same direction in ipsilateral fore- and hindlimbs, but it was much smaller in magnitude in hindlimbs (Lammers and Gauntner, 2008). Therefore, in forelimbs, the shear component of mediolateral force contributes more to shear force than in the hindlimbs because of limb contact location(Fig. 6). Theτ _CC,musc contributes only to shear force, and it is greater in the forelimbs. It appears that for the most part, theτ _CC,musc and shear component of mediolateral force within each limb pair cancel each other out. Third, the craniocaudal force,all of which contributes to shear force, apparently makes a minor contribution to the total shear force. Craniocaudal shear force differs from the other components of shear force because it presumably does not contribute to making the animal slip from the sides of the branch. Lammers and Biknevicius found that on a similar 2 cm arboreal trackway, M. domestica exerted significantly greater braking and propulsive impulses than the hindlimbs(Lammers and Biknevicius,2004); recently collected (A.R.L., unpublished) data confirm that pattern. It seems likely that craniocaudal force is not large enough to influence shear force so much that significant differences between fore- and hindlimbs are observed. Finally, it is also possible that there is some interaction between grip location and claw use or in the positioning of the digits. The opossums have claws on all digits except for the hallux, and the foot may more effectively grip the substrate and reduce shear force with claws or the somewhat opposable hallux.

The opossums did not change grip location on the 2 cm-diameter arboreal trackways to cope with the relatively smooth surface. Not only did θ not differ significantly between substrates, but F_shear was also not correlated with θ, or was very weakly correlated. [It is worth pointing out that τ_CC,musc contributes considerably to F_shear, and τ_CC,musc apparently has no relationship with θ (Lammers and Gauntner, 2008).] Correlations between peak and median F_normal and θ within each group (e.g. forelimb on the rough trackway) were usually weak or non-existent(r²≤0.32). Thus, it appears at first that grip location has little influence on how the opossums generate normal and shear forces. However, it is also possible that the within-group variation in θ is not great enough to produce strong correlations with normal and shear forces. The opossums may be biomechanically constrained to grip the 2 cm-diameter branch trackway such that the forelimb grips closer to the top of the branch and the hindlimb grips approximately midway between the top and the sides of the branch. If, on the smooth trackway, they place their hands or feet closer to the top of the branch, then normal force will greatly increase because a much greater proportion of the animal's weight is applied normal to the branch surface. But gripping the top of the branch provides very little bracing against mediolateral undulation or (in a natural environment) wind and branch movement because the opossum's hand has no opposable digits, and it seems likely that neither hand nor foot has a large enough span on the 2 cm branch to grip strongly. If the opossums grip the sides of the branch, then with sufficient mediolateral forces they could create a very stable grip with opposable limbs. But gripping the sides of the branch will increase the shear force because most of the vertical force will be tangential to the surface of the branch. Thus, it appears that the size of the branch and the sizes and morphologies of the hand and foot limit the ways that these opossums can adjust to a more slippery substrate.

The gray short-tailed opossums, with their morphology being relatively unspecialized for arboreal locomotion, seem constrained to grip the 2 cm branch in one particular way. When they attempt to move on a trackway with the same diameter but a smoother surface texture, the only way they can adapt is to move more slowly and increase the total substrate–limb contact time with as many limbs as possible. Many arboreal specialists have morphological differences that almost certainly allow greater flexibility in how they can move on rough or smooth branches. For example, many arboreal marsupials have significantly longer digits than closely related terrestrial marsupials(Lemelin, 1999). These longer digits probably allow the animals to grip branches more strongly than they can if their digits were shorter. Some arboreal rodents have pads with glands that secrete fluid so as to increase the friction between autopodia and substrate(Haffner, 1998). I predict that animals that are morphologically specialized to move in trees will have greater flexibility in terms of where and how they grip branches. An animal with long, opposable digits (or fluid-secreting glands, claws or microscopic setae that interact with the substrate at a molecular level)(Cartmill, 1985; Haffner, 1998; Autumn et al., 2006) should be able to grip the top of a branch, which means that much more of its body weight is applied normal to the branch. Thus, even when moving about on a relatively smooth or slippery branch, such a specialized animal will be able to avoid slowing its speed and adjusting its footfall patterns (as measured by duty factor and limb phase) as much as M. domestica.

LIST OF ABBREVIATIONS

 
• F_CC

  craniocaudal (or anteroposterior) force

 
• F_ML

  mediolateral force

 
• F_ML,normal

  normal component of mediolateral force

 
• F_ML,shear

  shear component of mediolateral force

 
• F_normal

  normal force (perpendicular to the substrate at the center of pressure)

 
• F_shear

  shear force (tangential to the substrate at the center of pressure)

 
• F_V

  vertical force

 
• F_V,normal

  normal component of vertical force

 
• F_V,shear

  shear component of vertical force

 
• SRF

  substrate reaction force

 
• θ

  the angle between a horizontal line intersecting the long axis of the branch and a line passing through the center of pressure of the limb and the long axis of the branch

 
• μreq

  required coefficient of friction (ratio of shear forces to normal forces)

 
• μs

  coefficient of static friction

 
• τCC,musc

  torque around the long axis of the branch, not resulting from the shear component of substrate reaction forces