R. McN. Alexander

The Gaits of Bipedal and Quadrupedal Animals

The gaits of reptiles, birds, and mammals are reviewed. It is shown that mammals of different sizes tend to move in dy namically similar fashion whenever their Froude numbers u2/gh are equal: here u is speed, g is the acceleration of free fall, and h is the height of the hip joint from the ground. The gaits of turtles and people are examined in detail. The gaits of turtles appear to reduce unwanted displacements (pitch, roll, etc.) to the minimum possible for animals with such slow muscles. The patterns of force exerted in human walking and running minimize the work required of the muscles at each speed. Much of the energy that would otherwise be neededfor running, by people and other large mammals, is saved by tendon elasticity.

A MINIMUM ENERGY COST HYPOTHESIS FOR HUMAN ARM TRAJECTORIES

Abstract. Many tasks require the arm to move from its initial position to a specified target position, but leave us free to choose the trajectory between them. This paper presents and tests the hypothesis that trajectories are chosen to minimize metabolic energy costs. Costs are calculated for the range of possible trajectories, for movements between the end points used in previously published experiments. Calculated energy minimizing trajectories for a model with biarticular elbow muscles agree well with observed trajectories for fast movements. Good agreement is also obtained for slow movements if they are assumed to be performed by slower muscles. A model in which all muscles are uniarticular is less successful in predicting observed trajectories. The effects of loads and of reversing the direction of movement are investigated.

Fourier analysis of forces exerted in walking and running.

Abstract The forces exerted on the ground by men walking and running have been recorded by means of a force platform. A simple method of Fourier analysis has been used to analyse the records. It shows differences between walking and running, between slow and fast walking, between accelerated and decelerated walking and between different individuals walking at the same speed.

Bipedal animals, and their differences from humans

Humans, birds and (occasionally) apes walk bipedally. Humans, birds, many lizards and (at their highest speeds) cockroaches run bipedally. Kangaroos, some rodents and many birds hop bipedally, and jerboas and crows use a skipping gait. This paper deals only with walking and running bipeds. Chimpanzees walk with their knees bent and their backs sloping forward. Most birds walk and run with their backs and femurs sloping at small angles to the horizontal, and with their knees bent. These differences from humans make meaningful comparisons of stride length, duty factor, etc., difficult, even with the aid of dimensionless parameters that would take account of size differences, if dynamic similarity were preserved. Lizards and cockroaches use wide trackways. Humans exert a two‐peaked pattern of force on the ground when walking, and an essentially single‐peaked pattern when running. The patterns of force exerted by apes and birds are never as markedly two‐peaked as in fast human walking. Comparisons with quadrupedal mammals of the same body mass show that human walking is relatively economical of metabolic energy, and human running is expensive. Bipedal locomotion is remarkably economical for wading birds, and expensive for geese and penguins.

Optimum timing of muscle activation for simple models of throwing.

In diverse throwing activities, muscles contract in sequence, starting with those furthest from the hand. This paper uses simple mathematical models, each with just two muscles, to investigate the consequences of this sequential contraction. One model was suggested by shot putting, another by underarm throwing and the third by overarm throwing, but all are much simpler than real human movements. In each case there is an optimum delay between activation of the more proximal muscle and of the more distal one, that maximizes the speed at which the missile leaves the hand. If the delay is shorter than optimal, the throw is completed sooner and less time is available for contraction of the proximal muscle: it may shorten faster, exerting less torque, or through less than its full range of movement, and so do less work. If it is longer than optimal, less time is available for contraction of the distal muscle, which therefore does less work. The optimal delay is in some cases longer than would maximize total work because the delay influences the proportion of the work that appears as kinetic energy of the missile.

Bending of cylindrical animals with helical fibres in their skin or cuticle

Many animals, including some nematode worms, fish and whales have crossed helical fibres in their skin or cuticle. Such fibres may stretch or shorten as the animal bends, depending on their arrangement. This is investigated by calculation for two models: one has the fibres free from any underlying skeleton (as in nematode worms) while the other has them attached to rigid vertebral processes (as in whales). The first model will not remain straight, with the fibres taut, but will tend to bend one way or the other. The second model tends to spring back to the straight position after being bent, if the fibres make angles greater than 60° with the longitudinal direction. Thus fibre elasticity can help to maintain swimming movements. However, if the fibre angle is 60°, the fibres keep constant length in bending and have no tendency to return the body to any particular position.

Optimum strengths for bones liable to fatigue and accidental fracture

Limb bones are liable to fail by fatigue, due to stresses imposed repeatedly in activities such as running. They may also be broken by larger stresses which occur occasionally in accidents. Too weak a bone will probably fail but too strong a bone is unduly heavy. A mathematical model predicts optimum strengths for bones subject to the hazards both of fatigue and of accidents. When accidents are mild and predictable, fatigue is the more important hazard. When they are large and unpredictable they are the more important hazard and stronger bones are generally preferred, but in extreme cases it may become advantageous to dispense with the bone. There is a restricted range of circumstances in which fatigue and accidents are both important hazards. The strengths of many limb bones seem surprisingly low, in the light of the theory and of current knowledge of the fatigue properties of bone.

Forces in animal joints

Estimates have been made of the forces which act in major joints of various arthropods and vertebrates, in strenuous activities. Over the range of size from fleas to dinosaurs, joint forces tend to be roughly proportional to (body mass)-1/3. The maximum stresses which occur in joints are probably of the same order of magnitude in mammals of all sizes and also in insects of all sizes, but the stresses in insects are probably larger than the stresses in mammals and other vertebrates.

The scope and aims of functional and ecological morphology

Functional morphology asks how animal structures work and whether observed structures are superior to possible alternatives. Ecological morphology examines the structures of different animals in relation to their various environments and ways of life. Some studies in both fields are concerned mainly with topography, but most make use of at least one of the branches of physics and engineering: examples involving kinematics, dynamics, fluid mechanics, acoustics, optics, heat exchange and diffusion are presented, together with some of the important experimental methods. Whichever branches of physics are used, allometry, optimization and the study of symmorphoses are likely to be illuminating. Functional and ecological morphologists should be more aware of evolutionary theory.

Elastic Mechanisms in the Locomotion of Vertebrates

Elastic mechanisms serve many functions in the locomotion of vertebrate animals. The ligamentum nuchae in the necks of ungulates gives some support to the head while allowing it to be lowered to the ground for feeding. Tendons in the distal parts of the legs of mammals save energy by stretching and recoiling in each step, enabling the animal to bounce along like a child on a pogo-stick. The same tendons aid jumping by recoiling rapidly like a catapult. An aponeurosis in the back serves as a spring in galloping, halting and reversing the swing of the legs. The elastic compliance of paw pads cushions impacts with the ground. The compliance of the tendons that move our fingers makes it easier for us to control the forces exerted by our fingers but harder to control their positions. The elastic properties of the diaphragm and abdominal wall may give resonant properties to the respiratory system, enabling running movements to drive breathing in some mammals. These and other possible functions of elastic mechanisms are reviewed.