A point-mass model of gibbon locomotion

The swinging motion seen when brachiators move beneath their overhead supports naturally brings to mind the oscillations of a pendulum and the repeated interchange of gravitational potential energy (E_p) with kinetic energy (E_k). It is tempting, then, to assume that brachiating apes use natural pendular motions to reduce the muscular investment necessary to travel within the forest canopy (Ashton and Oxnard, 1964a,b; Fleagle, 1974; Preuschoft and Demes, 1984). Previous pendular swinging models are all based explicitly or implicitly on a half cycle of a freely swinging pendulum.

Under some circumstances, brachiators do move roughly according to the predictions of simple oscillatory pendular motion. Preuschoft and Demes (1984) modeled brachiation as a distributed-mass simple pendulum. They found good agreement with observed brachiation in gibbons for relatively slow brachiation. They predicted a natural pendular period of 0.98 s, while empirically measured periods in similar sized gibbons ranged from 0.83 to 0.95 s for slow, continuous-contact brachiation. Some limitations in the pendular swinging models have been attributed to model oversimplification with regard to both poor representation of the complex mass distribution of the animal and potential movements of non-support limbs during the swing (Fleagle, 1974; Preuschoft and Demes, 1984). Swartz (1989) pointed to the narrow range in oscillation frequency that pendular swinging models allow. Preuschoft and Demes (1984) recognized many of the features affecting ricochetal brachiation (see their Fig. 12.5), but the pendular model they presented did not calculate any predictions for the more athletic ricochetal brachiation maneuvers of gibbons.

Gibbons display two brachiation gaits. At slower rates of forward progression, the gibbon uses a continuous-contact gait that is characterized by a dual-limbed support phase during a stride cycle. As in human walking, the duty factor of each limb is ⩾0.5 and there is always at least one limb in contact with the overhead support. At higher forward velocities the animal uses a ricochetal brachiation gait that is characterized by a non-contact phase between support phases of each step (the step is defined here as the period from hand contact to the next contralateral hand contact).

Here we describe a simple model which extends the pendular continuous-contact models in a simple way for gibbon ricochetal brachiation. With this mechanism, step periods can be arbitrarily long or short, even with a given pendulum length.

In the simplest rendition of the pendular swinging model, brachiation is viewed as a sequence of half-cycle motions of a simple point-mass pendulum. In this model, the center of mass moves along a sequence of circular arcs (Fig. 1A). If friction and elasticity are neglected, the system can continue its motion without investment of energy above that necessary to place the next supporting arm into the appropriate position for the following step. In this system, no energy is lost in the interactions of the system with its support because collisional losses are eliminated by the motion coming to a rest at the end of each arc. Hence the swing period is a half cycle of a simple pendulum. Higher frequency motions are imaginable but require energy supply to make up for collisional losses at the discontinuity as the center of mass changes direction at the transition from one circular arc to the next. In this paper, the only continuous-contact solutions we consider are those in which the center of mass (CM) comes to rest at the top of the swing when the change of hand holds occurs.

Like the ricochetal gait of a brachiator, the ricochetal model we propose utilizes a flight phase between hand contacts. We take the motion of the ricochetal brachiator to be a circular motion around the hand hold at full arm extension during support, combined with a parabolic free-flight arc between hand holds (Fig. 1B). We assume that the animal can coordinate its arms to have its hands in the proper place for each landing into a new circular arc around the next hand hold (limb mass is neglected in the model). In this model, collisional losses can again be reduced to zero by assuming that the gibbon chooses motions that make a smooth transition from the parabolic flight phase to the circular swing-support phase of the stride. That is, mechanically work-free motion of this model is possible if the tangent of the flight path during the flight portion of the stride is coincident with the arc of the swing as the animal begins the support portion of the stride (Fig. 1B, inset). Although this model appears to have a rather smooth motion, it does predict a discontinuity in acceleration and handhold reaction force when the ballistic flight phase begins and ends.

If the handhold spacing is less than twice the arm length, then both the continuous contact and ricochetal modes of mechanically cost-free transport are available. These models, together, are essentially equivalent to the minimal-biped model of Alexander (1995) turned upside down. These models cannot quite be classified with the passive-dynamic locomotion models of the types recently investigated for walking (e.g. McGeer, 1990; Garcia et al., 1998) because, as for the swinging leg in Alexander’s minimal-biped model, the non-contacting arm in this minimal-bimanual model is assumed to be appropriately coordinated by a conscious (non-passive) controller. This two-dimensional point-mass model is probably the simplest model that could feasibly provide useful information about the dynamics of gibbon brachiation.

The brachiator is modeled as a massless inextensible link with length L connected to a point mass m. As a guess for comparison with data, L represents the distance from the animal’s CM to the hand grip when hanging (this definition might slightly contradict another possibly useful definition of L as half the distance between the most distant objects the animal can simultaneously hold). We want to track the position of the center of mass (x, y) in time t as the animal moves from one hand hold to the next, a horizontal distance D away. While swinging, the pendulum makes an angle ϕ with the horizontal until the initiation of flight at ϕ_i. The speed v has a maximum speed v_b at the bottom of pendular swing arc when the tension in the pendulum is also at its maximum T_b. The speed at the initiation of flight is v_i, and the average horizontal speed is v_av=D/t_tot where t_tot=t_c+t_f and t_c and t_f are the pendular contact time and the flight time, respectively. The total, potential and kinetic energies are E, E_p and E_k, respectively. We denote differentiation with respect to time by superposed dots.

For most calculations, we assume a value for the handhold spacing D and takeoff angle ϕ_i and calculate other quantities of interest as follows. Since the flight time is t_f=2[(v_isinϕ_i)/g] and the horizontal component of flight velocity is v_icosϕ_i,

For a given handhold spacing there are a number of different solutions of the center of mass trajectories, as shown to scale in Fig. 2, in this case with D=4L. In Figs 3A–D, 4A, various motion quantities are shown for families of solutions. The extremes of these solutions are obviously beyond the capabilities of an animal for a variety of reasons. From these solutions we can make the following observations. (1) For a given handhold spacing greater than 2L, there are high energy ricochetal solutions for both low and high flight initiation angles (Figs 2, 3C). (2) At 2Lsinϕ_i=D, the ricochetal solution matches the continuous-contact swinging solution. (See Fig. 3A, for example.) (3) For handhold spacings of less than 2L, there are ricochetal solutions only so long as 2Lsinϕ_i<D. In Fig. 3A, for example, the solutions at a given D<2L have a maximum ϕ_i where they meet the continuous-contact solution. (4) For D<2L contact time t_c is an increasing function of flight initiation angle, but is always less than the period of a simple pendulum with the same amplitude motion (i.e. the continuous-contact solution). (5) For all but the nearly continuous-contact motions, the contact time in ricochetal motion is less than the period of a linearized simple pendulum (π in the dimensionless plot of Fig. 3A). But the flight phase is arbitrarily long, so there are no fundamental restrictions on stride period for a given hand spacing greater than 2L (see Fig. 3B). (6) The average forward speed is seen to be highest at small flight initiation angles where the flights are more flat (Fig. 3D).

In order to reduce the range of possible solutions that satisfy our general criteria (i.e. tangency of the swing arc and flight parabola), one would like to define an optimization criterion. The most obvious optimization criterion is the specific energetic cost of transport (energy used per unit weight per unit distance). However, this measure is already optimal (at 0) for all the solutions shown. None of the motions considered require any energy input, at least theoretically. Another candidate for optimization is speed; however, the model allows arbitrarily high speeds (the low trajectories in Fig. 2, corresponding to small ϕ_i in Fig. 3D, are the fast ones). Thus, we need to seek a more subtle candidate for the optimization. For an animal that does not want to waste energy starting, stopping or changing gaits, we might seek minimum total energy solutions. Note that, in Fig. 2, the low, fast solutions (small ϕ_i) have high total energy as do the high, looping (large ϕ_i) solutions (see Fig. 3C) for D>2L. Also, it is likely to be advantagous for an animal to reduce the force demands on its muscles or bones or to reduce stress. As noted, the solution that minimizes the total energy of the system is also the one that minimizes the maximum tensile force in the supporting arm. Thus, we can tentatively make more definitive predictions about the model by picking an ‘optimal’ motion from amongst the range of possible motions. Fig. 3C shows the total energy of some solutions at various values of D, as a function of the take-off angle ϕ_i. As can be seen in Fig. 3C and can be shown by differentiation of the equations above, the optimal solution involves a flight initiation angle of ϕ_i=45 ° (so long as ⁠). This choice of ϕ_i agrees with the classical ballistics intuition that the greatest range is achieved when a frictionless cannonball is launched at 45 °. [The set of parabolic trajectories of a fixed energy whose apex is on the y axis have an envelope made of two sloping 45 ° sloping angles. Finding the lowest energy trajectory is equivalent to sliding the apex of this wedge up to the lowest point where the envelope matches the boundary conditions, i.e. either the launch and land points (from classical ballistics) or the launch and land circles (from the ricochetal oscillation model).]

Even though the mechanical energy is minimized by a flight initiation angle of 45 °, the total mechanical energy of the model is within 20 % of this optimum for flight initiation angles from 32 ° to 62 °. Thus, we cannot treat the optimization criterion as one that would tightly constrain behavior, given that other quantities we have neglected likely need to be controlled and optimized by an animal as well.

The primary metabolic cost of locomotion may be for turning on and off muscle force and not for doing work, as proposed for running by Kram and Taylor (1990). If the metabolic cost of gaining a momentary isometric tension is proportional to the magnitude of tension in the support limb and roughly independent of time, the metabolic cost per unit distance of transport would be proportional to peak tension divided by the distance between contractions D. This measure of ‘cost’ (see Fig. 4A) also has a minimum at ϕ_i=45 °. This minimum is also broad so the cost function is nearly optimized for a large range of flight initiation angles. Surprisingly, this cost is almost independent of handhold spacing (the curves in Fig. 4A nearly fall on top of each other).

Finally, if we accept ϕ_i=45 ° as the preferred ricochetal mode, since it minimizes energy, peak arm tension and ‘cost’, we can look at various measures of performance for continuous-contact motion and for ‘optimal’ ricochetal motion, as shown in Fig. 4B. Shown are peak tension (the total energy curve would be similar), average speed and the ‘cost’ function described above. Tension and energy are minimized by arbitrarily small handhold spacings but average speed and ‘cost’ both favor arbitrarily large handhold spacings.

Limited reach alone, as determined by extended arm length, demands a transition to ricochetal motion for handhold spacings D>2L. But ricochetal motions are possible for arbitrarily small handhold spacings. The three optimal conditions discussed all predict that the (barely) ricochetal motion corresponding to continuous-contact motion is best for handhold spacings less than and that ricochetal solutions are preferred for larger spacings. In Fig. 4B, the continuous-contact solutions are shown within the band that corresponds to ⁠, even though they are not optimal there. The figure makes evident that a gibbon would suffer little penalty for any of the three optimal criteria for delaying that transition until at least about 1.7L. Such a gait-transition delay would postpone the obvious decrease in security and maneuverability caused by losing hand contact.

In order to determine how well the model described above, either restricted by the optimization criterion or not, corresponds to the brachiation behavior of a gibbon, we compared some of the predictions of the model with measurements of the brachiation in a 7.95 kg female White-handed gibbon (Hylobates lar). The techniques utilized to make these measurements are fully described in Chang et al. (1997). Briefly, a universal force transducer of our own design was mounted to the ceiling of a reinforced chain-link exercise cage of dimensions 3.71 m wide, 6.10 m long and 3.56 m high. Uninstrumented hand holds identical to that of the transducer were also mounted to the ceiling of the exercise cage in series with the transducer hand hold. The distance between hand holds for different analysis sessions was adjusted to be at equal intervals of either 0.8, 1.2, 1.6, 1.72, 1.95 or 2.25 m. The number of hand holds was maximum at seven (the instrumented transducer in the center of six non-instrumented hand holds) and minimum at three (the transducer flanked by two non-instrumented hand holds). The cage restricted the number of hand holds for longer spacings; at these longer distances the animal always used ricochetal brachiation. Only those runs in which the change in horizontal velocity was small were used in this analysis, as determined by integration of the horizontal load. A lateral video image was taken (60 Hz, 1/500 s shutter speed) as the animal brachiated freely across the hand holds.

Time of contact with the hand hold for a given run was determined from the force transducer output. Contact with the transducer was easily determined and was accurate to 5 ms, which was the channel sampling rate. Maximum and initial velocities were determined by integration of the force record after conversion to acceleration. Average forward velocity was determined from the calibrated video image. The video images of each run were digitized and the time and distance interval of a complete brachiation cycle was determined (NIH Image). This velocity value was used to determine the constant of integration for the horizontal motion from load cell data. The constant was assumed zero in the vertical direction because the hand holds were all at the same height.

Fig. 5 shows the relationship between average forward velocity and handhold spacing, comparing a brachiating gibbon with the model predictions. The model predicts two ‘gaits’: continuous-contact and ricochetal. At slower forward velocities and closer handhold spacing, the optimization criterion selects continuous-contact gait, in which at least one hand is in contact with the hand hold at all times. The model is then just a sequence of simple pendulum motions placed side by side. At higher forward velocities and for handhold spacings with ⁠, the optimization criterion selects the ricochetal solution with a flight initiation angle of 45 ° (the theoretical portions of Fig. 5 duplicate the v_ave curve in Fig. 4B).

The time course of the vertical and horizontal forces predicted by the model and measured in a brachiating gibbon is illustrated in Fig. 6. The fluctuations of force are much smoother in the gibbon compared with the model. The model underestimates the peak vertical force, although the net impulse is similar.

There is substantial overlap between the prediction of the relationship between v_ave and handhold spacing of the model and the behavior observed in the brachiating gibbon (Fig. 5). The changeover from continuous-contact to ricochetal brachiation corresponds well between the model and observations. The model tended to overestimate v_ave except at the longest spacing of the continuous-contact gait (which corresponds to the changeover point of the two gaits) and the longer spacings of the ricochetal gait (1.9 and 2.25 m). Although the observed velocities do not differ significantly from the model’s predictions (i.e. the model prediction lies within the 95 % confidence interval of velocities measured at each of the spacings), they are not adequate to fully evaluate the predictive capability of this model.

Although the theoretical prediction is as smooth as the model allows, the gibbon chooses an even smoother force profile (Fig. 6). This observation reinforces the notion that the avoidance of energy-dissipating inelastic collisions may be a key behavioral feature of brachiation. Smoothness of force transition is a way of further avoiding collision losses. The origin of the word ‘jerk’ for the time derivative of acceleration is probably due to the fact that, in systems with play or flexibility, a large rate of acceleration leads to internal collisions or internal motions. The gibbon, being actually more than a point mass, may well be avoiding such ‘jerk’ losses by maintaining a smooth force profile.

The simple pendulum and the ricochetal model motions we describe have some basic differences. This is the case even though they both swing below their support and the simple pendulum represents a subset of the ricochetal solutions. One demonstration of this difference is indicated by the relationship between initial velocity of the body when it contacts the support and the maximum velocity it achieves during the swing. The simple pendulum model begins at an initial velocity of zero. The greater the arc of a pendulum (i.e. the larger the angle it swings through), the higher the velocity it will achieve at the bottom of the swing. Thus, for continuous contact, as support spacing or v_ave increases, the greater the difference between the v_i and v_b. This is in contrast to the ricochetal case. The ricochetal model makes contact with the support with a substantial v_i. The farther the handhold spacing or the faster the animal travels, the smaller the difference between v_i and v_b. In Fig. 7, the difference between v_b and v_i is plotted against v_ave for the pendulum and ricochetal models and for the gibbon brachiating in a continuous contact and ricochetal manner. Compared to the data, the simple point-mass model generally underestimates the velocity difference. However, continuous-contact brachiation shows a distinctive increasing trend that parallels the pendular prediction while ricochetal brachiation shows a decreasing trend that parallels the ricochetal prediction.

Time of hand contact (t_c) is a critical determinant of the interaction between the animal and its support. The time of contact is a measure of the time available for the limb’s support musculature to be active and has been shown to correlate well with metabolic cost in terrestrial runners (Kram and Taylor, 1990). A plot of t_c against v_b has two distinctive predictions for the pendulum versus ricochetal motions (Fig. 7B). The maximum velocities v_b of the two models overlap substantially. The continuous-contact brachiation runs are scattered generally around the pendular model prediction, but the fit is not good. The behavior of the animal does not appear to be strongly influenced by the optimal pendular dynamics in continuous contact. However, the ricochetal runs are tightly grouped around the ricochetal prediction for these variables (Fig. 7B).

The mechanics of brachiation have been conceptualized in the past exclusively as a simple pendulum (Andrews and Groves, 1976; Ashton and Oxnard, 1964a,b; Avis, 1962; Carpenter and Durham, 1969; Chivers, 1974; Fleagle, 1974; Jenkins et al., 1978; Jungers and Stern, 1984; Mittermeier, 1978; Parsons and Taylor, 1977; Preuschoft and Demes, 1984). When the pendulum model did not fulfill expectations, it was assumed that the details of the model did not adequately describe the complexities of the animal and its limbs, rather than questioning whether the pendular mechanism was an appropriate or adequate model of the behavior of these animals (Swartz, 1989). Our gibbon data indicate that only under limited circumstances is the movement of the gibbon determined by continuous-contact pendular mechanics. Even for continuous contact, the gibbon does not appear to be strongly restricted to the simple pendulum-like motions (Fig. 7).

Many aspects of our basic model match the model described by Preuschoft and Demes (1984), and it is interesting that many of their observations correspond with the quantitative results of our model. Preuschoft and Demes described the condition at the transition from the flight phase to the contact phase of the ricochetal stride that corresponds to the non-collision criterion we define. However, they did not define this condition as a critical criterion of ricochetal brachiation. They also did not identify optimization criteria or extend their model to quantify predicted ricochetal behavior.

Gibbons use ricochetal oscillations as a mechanism of bouncing between overhead hand holds in their higher velocity brachiation movements. Our simple model offers a solution to the frequency restrictions associated with continuous-contact pendulum-based mechanisms. At the same time, the model offers a mechanism for minimizing energetic cost in brachiation. Theoretically, the ricochetal model is 100 % efficient, just as the continuous-contact pendular exchange and some models of running with springs.

In order to interpret the specialized morphological adaptations that characterize the brachiators, it is first necessary to determine the mechanical function(s) that their specific morphology allows. One major confusion that has obscured the interpretation of the anatomical specializations found in brachiators (limb and body proportions, joint anatomy, muscular activity and coordination, etc.) has been the misinterpretation of brachiation as primarily a pendulum-like movement. The data provided in this study show some evidence that, for ricochetal brachiation, the point-mass ricochetal model approximates both the scaling and trends in dynamic changes in locomotion. This is in spite of the model’s gross oversimplification of the animal’s complex morphology. It must be recognized, however, that in the ricochetal gait the gibbon will not behave as a simple point-mass pendulum. Instead, the animal acts more like a jointed complex pendulum (J. E. A. Bertram and Y. H. Chang, submitted for publication). This difference likely accounts for some of the differences in the model’s predictions and our direct measurements of gibbon brachiation (Figs 6, 7). Nevertheless, the trends described by our simple model clearly indicate that the gibbon is moving very much as we predict.

A feature of this model is the wide range of motions that it allows at near ‘optimal’ speeds. Gibbons appear to have developed a very simple, yet sophisticated strategy that allows them to maneuver at high velocity and change the distance between ‘steps’ with minimal energy loss. It is remarkably difficult for most terrestrial animals to arbitrarily alter the length of successive steps when running at high velocities, yet this is what is routinely demanded of the brachiator in its natural environment. Our model allows for a wide range of step lengths. Strategies based on this required functional flexibility may account for why brachiation is not such an economical mode of locomotion as our simplistic theory indicates it could be. For instance, based on analysis of oxygen consumption rates, it has been found in the neotropical spider monkeys Ateles geoffroyi and A. belzebuth that the cost of locomotion was always greater in brachiation versus quadrupedal walking (Parsons and Taylor, 1977).

Further analysis of the mechanics underlying brachiation will provide better understanding of the morphological specializations that have allowed brachiators to fully exploit their three-dimensionally complex environment. Brachiation may prove to be a fruitful area to investigate the quantitative relationship between functional morphospace and the adaptation of an integrated system to a complex locomotory task.

The animals and facilities necessary to quantify gibbon brachiation were provided by the Department of Anatomical Sciences, SUNY-Stony Brook. We are particularly indebted to Susan Larson, Jack Stern, Brigitta Demes and Marianne Crisci for assistance with this project. We also wish to thank Mariano Garcia for assistance with modeling concepts. Some of the work reported here was supported by the NASA Space Grant College and Fellowship Program, NASA/NY Space Grant Consortium (NASA NGT 40012). The brachiation analyses described in this paper were supported by a grant from NSF-Division of Physical Anthropology (NSF-SBR-9422118 and 9706225).