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Understanding brachiation: insight from a collisional perspective

James R. Usherwood and John E. A. Bertram^*

Food, Nutrition and Exercise Sciences, Sandels Building, Florida State University, Tallahassee, Florida 32306, USA

View larger version (14K): [in a new window]   Fig. 1. The path of a point mass (filled circles) with a support able to oppose forces only in tension as it moves past a rigid handhold. The path of the mass prior to collision is indicated by the broken line; the straight, rigid mass-handhold connection at the moment of collision by the bold line; the velocity vectors relating to the velocity of the mass an instant before (V) and after (V') collision, by the straight arrows; and the arc described by the handhold-centre of mass length L about the handhold as a large circle.

View larger version (23K): [in a new window]   Fig. 2. Parameters required for Model 1 demonstrated using kinematics of Example Run B. The kinematics (A) show tracings of a brachiating gibbon derived from video recordings at 0.083 s intervals. Blue outlines indicate contact with the first handhold, green, contact with both handholds and red, contact with the second handhold. The centre of mass (CoM)-handhold distance L is taken from the bottom of the first swing (B). A second value of L relating to the CoM to handhold distance at the bottom of the second swing is shown as L', and its consequences are displayed in Fig. 5. (C) Two potential release angles, 1 and 2, and their resulting ballistic paths. The grey arc displays the result of release at the earlier, lower angle, 1, which results in a narrow miss of the contact circle. At the later, higher release angle, 2, the contact circle is overlapped. Between these two release angles, and their subsequent arcs, lies a strategy that results in `ideal' contact, with path-matching resulting in zero collision loss. Such ideal paths are highlighted in yellow throughout.

View larger version (14K): [in a new window]   Fig. 3. Geometry for Model 2, a point-mass model for the energy loss due to collision for a range of overshoots e and handhold-centre of mass (CoM) lengths L. A point-CoM and an extended, rigid connection to the hand, are shown. The path prior to collision is indicated by a broken line. (i) The geometry of collision for short (A) and long (B) arms at ideal, no-loss contact, (ii) before collision but with an overshoot, and (iii) at the moment of collision, given an overshoot e. Collision angles (ß) related to contact with shorter handhold-CoM lengths (both indicated with subscript A) are smaller than those related to contact with longer handhold-CoM lengths (subscript B).

View larger version (27K): [in a new window]   Fig. 4. Model 1 for simple continuous-contact brachiation with D=1.2 m. (A) Kinematics show tracings derived from video recordings at 0.083 s intervals. Blue outlines indicate contact with the first handhold, green, contact with both handholds, and red, contact with the second handhold. (B) The range of ballistic paths relating to =0° to =90°, given the observed brachiation energy. Each arc represents a separate trajectory due to differing angles of release: steeper arcs relate to later (higher release angle) releases, with lower release velocities due to the conversion of kinetic energy into potential through the swing. Arrows highlight the final direction of selected trajectories. The potential release angles can be put into three groups: (i) `early release', leading to a path that does not intersect with an arc described by L about the second handhold (i.e. a fall); (ii) `adequate release' resulting in ballistic paths that allow contact with the second handhold, albeit with some collision energy loss; and (iii) those angles that cannot be achieved because the required potential energy would be greater than the total mechanical energy of the gibbon. The underlying yellow regions (B,D) denotes the `ideal', zero-collision-loss strategy. The underlying green shading (B,D) denotes the consequences of release at the maximum height possible for the observed energy. (C) Tracings before (blue), at (green) and beyond (red) double contact are aligned vertically to show the backwards movement of the centre of mass at the top of the swing. Broken lines indicate the approximate position of the centre of mass, and suggest a `loop-the-loop' path due in part to the active flexion of the trailing arm. (D) The energetic consequences of collision are shown in alternative forms. The y-axes show energy loss in absolute terms, or proportional to `brachiation energy', which relate directly. The x-axes show the release conditions, either in terms of release angle (bottom) or time of release (top). The x-axes, however, do not relate directly: there is a greater time difference between angles of release at higher angles, as the mass moves more slowly, due to conversion of kinetic to potential energy. Energy loss relating to `angle at release' is shown in black, that relating to `time of release' is indicated in dark red. Regions marked (i)–(iii) correspond in B and D.

View larger version (26K): [in a new window]   Fig. 5. Results of Model 1 for the kinematics shown in Fig. 2A (D=1.2 m), of continuous-contact brachiation, demonstrating the benefits of raising the centre of mass with a leg-lift. The potential ballistic paths are shown (A,C) and energetic consequences of collision (B,D) are derived from Model 1, with L' replacing L as the sole difference for plots C and D. Both potential ballistic paths and energetic consequences of collision are qualitatively similar to those described for Fig. 4. The leg-lift considerably reduces energetic collision losses. Axes and underlying highlights follow the notation in Fig. 4.

View larger version (38K): [in a new window]   Fig. 6. Results of Model 1 for ricochetal brachiation (D=1.93 m). Potential ballistic paths (A–C) are separated into three groups: Early release (A), which would result in paths under-shooting the second CoM-handhold arc; Successful release (B), which would allow successful contact to be made to the second handhold; and Late release (C), showing an alternative range of paths that would result in a fall. Arrows highlight the direction of selected trajectories. Tracings in (D) are as described for Fig. 4A, except that green outlines relate to instances when neither hand is in contact with a handhold. The energetic consequences of collision (E) are presented as for Fig. 4D. The underlying yellow regions relate to the two potential paths entailing zero collision loss, also highlighted in B. Selected tracings of the second swing (F) emphasise: (i) initial contact; (ii) `overshoot', with flexed support arm; and (iii) three tracings displaying the `double-pendulum' action as the hips revolve about the shoulders prior to the bottom of the swing.

View larger version (18K): [in a new window]   Fig. 7. Results of Model 2. Proportional energy loss due to collision with hand to centre of mass length L ranging from 0.3 to 1.1 m and overshoot e from 0 to 0.3 m. Negative e (undershoots) would result in missing the handhold, and a fall. The observed gibbon morphology relates most closely to L=0.7, indicated by the bold line.

View larger version (17K): [in a new window]   Fig. 8. Body posture without (A) and with (B) second moment of mass about the handhold maximised; a development from Fig. 3. The straight broken lines indicate the path prior to collision. With no overshoot (i), body orientation is unimportant and collision losses can be avoided. Given an overshoot e (ii, during which the handhold may be held lightly, and iii, the instant of collision), the posture with the body aligned perpendicular to the path (B) has a greater second moment of mass, and the overshoot results in smaller loss of energy due to collision. A consequence of aligning the body perpendicular to the path is the `double pendulum' action (Biv), with rotation of the body about the shoulder (dotted line), which is commonly observed in ricochetal brachiation (see Fig. 6).