<| Dynamic Forces

Vertical Forces. The first two figures show the forces acting upon the legs of running humans, trotting dogs, and running roaches during ground impact. Vector lengths relate to direction and magnitude of the forces at equal time intervals during the stance [ground contact phase]. These diagrams come from the work of R.J. Full and his associates [Dick00]. Because of their rotated-under-the-body legs and upright stances, the forces in the human and dog point "through" the hip or shoulder of each leg, minimizing the torque at each joint. In each step, there is an initial deceleration phase, where the muscles and tendons absorb the impact forces, followed later by an acceleration phase, as the legs push off into the next step. Self-Stabilization. In contrast, the roach has a sprawled stance with the legs spread. During running, the front legs decelerate backwards during ground impact to absorb forces, while the hind legs push forward to accelerate the animal into the next step. The middle legs do both. They push backwards during ground impact, and then forwards during the next step. The roach uses an alternating tripod gait, where a tripod consists of front-back legs on one side and middle leg on the other. Regards ground impact forces during a step, each tripod operates similar to 1 leg of a human [see below]. What is very different from the human and dog, however, is that the lateral spread of the roach's legs produces significant "sideways" forces and torques on the body. In normal operation, all 6 legs tend to push the body towards the mid-line. Thus, the explicit anatomy of the roach leads to automatic lateral self-stabilization during locomotion [Kubr99]. The roach can partially compensate for sideways forces by bending its legs on impact and then springing back, but unless a human "leans into" a large sideways force, he will likely be bowled over. The roach's sprawled posture produces a first level of compensation missing in the human. Just try to fip a roach. Ground Impact. The diagram on the right shows the similarity of ground impact forces versus time for a range of animals - from [Full00]. Each bump represents one step [half the total cycle]. One or more legs work together to distribute the forces over space and time during each step. In the roach, for instance, each bump represents one tripod step. The time phasing of the 3 tripod legs acting upon the ground - front leg decelerating, rear accelerating, and middle leg doing both - results in the integrated curve shown. In the dog, each bump is the integrated force on a "diagonal" [opposite corner legs]. These measurements, plus the considerations mentioned above, have led R.J. Full to conclude that, across the animal kingdom, legs and locomotion work in an analogous fashion - namely, "... 1 human leg works like 2 dog legs, 3 cockroach legs and 4 crab legs ...". Raibert found a similar result relating quadruped gaits to a virtual biped gait [Raib84]. This makes some sense, of course, since these animals are all bilaterally symmetrical around the midline. Note that, because of their sprawled postures, 6-leggers do have relatively large lateral forces compared to the vertical - related to the self-stabilization discussed above. Also, during running, humans and dogs will go completely airborne for some period of time, while the hexapods tend to bounce like a pogo stick without lifting off.

<| Energy Considerations

Inverted Pendulums. There are several simplified models to describe leg dynamics. The simplest treats the body as an inverted pendulum mass which transforms energy back and forth from gravitational potential energy at the top of the stance phase to kinetic energy during the lift phase of the step. In this model, the leg and body essentially rotate around the downed foot as a pivot point, with the up and down motions of the body mass related to the energy transformations. SLIP. The IP model has some credibility for walking animals but is not very appropriate for running animals, which also store significant energy in the flexing of muscles and elastic tendons, so the more realistic SLIP [spring-loaded inverted pendulum] or spring-mass model has been developed. In the SLIP model, besides potential-kinetic energy transformations in the mass, energy is also stored in the springs during the deceleration [impact] phase of the step, and then regained during the acceleration [propulsion] phase. The low point in the S-M diagram corresponds to the middle of the ground impact phase of the running leg, when it is maximally bent and energy has been stored in the flexing of the muscles and tension in the tendons. [Full00] presents an excellent discussion of the spring-mass model. In real animals, energy recovery can be upwards to 70% efficient. Horse tendons. Related to this, the diagram at the right shows the muscle (dark colors) and tendon / ligament (light colors) arrangement on a horse. Note that main muscle mass is located near the torso of the animal, and there is mainly tendon with little muscle in the lowest leg segments. These segments are moved, either by muscles pulling on tendon and bone above, or by joints bending and tendons "stretching" in reaction to ground impact forces. During the stride when a leg impacts the ground, the tendons stretch to take up ground shock and store energy which is recovered during push off into the follow-on step. In the front legs, most of the impact is taken up by bending and lowering of the fetlocks (horse, point 15), while it is absorbed by bending at the hocks (horse, point 7) in the rear legs. For leg bending comparisons, see horse, and compare to dog. The front knees (horse, point 14) do not bend backwards, and in both dog and horse they come down straight, so the impacts are mainly absorbed in the feet. The large flexor tendons on the rear side of the fetlocks of the horse (obvious in the diagram) are the main stretch elements in the lower legs. Their form and placement, as related to the angle of the fetlock, clearly indicates their function.

<| Nature's Roadster

Cockroaches are the speedsters of the arthropod set, having been clocked at upwards to 1.5 m/s, or 3.4 MPH. At slow speeds, they walk on 6 legs, but as their speed increases, they go to 4 and eventually 2, and afterwards even become airborne part of the time - topping out at up to 50 times their body length per second [Full91]. On the right below - a roach doing a 4-legged, alternating diagonal trot. When doing this, they use many forms of feedback, including equilibrium [inner ear], vision, strain, wind, touch, and "visco-elastic", as well as many forms of control, including neuronal, skeletomuscular, inertial, and potential energy. Their basically "self-stabilizing" anatomy is operated upon by their nervous control structures, which in turn are modulated by all forms of real-time sensory feedback. Furthermore, this is all very successful because roaches have been around for 100s of millions of years, and during this time have been difficult, at best, to apprehend and incarcerate successfully. Roaches are also unhappy house guests. What food they do not wish to eat, they foul on their way out, as a thank you note. The following is a list of some of the sensors that go into animal behavior: external: eyes, ears, noses. equilibrium organs: inner ears, statocysts, halteres [wing stubs used by flies as gyros]. mechano-sensory: contact, pain, wind, proprioceptive, strain, temperature. The following is a list of some of the effectors that go into animal behavior: skeletal: external skeleton, muscle, tendon. neural: brain, sensory nerves, control nerves, local reflex arcs. postural: leg anatomy, body geometry, posture, visco-eleastic mechanical reflexes, When running, the roach is able to pull all of these aspects together simultaneously, and blow right on by any other 6-legger in the neighborhood. On 6 legs, the roach moves tripod-to-tripod like other insects. Faster, on 4 legs, it moves similar to a running dog or horse. Down to 2 legs, it steps like a bird or human. In these cases, the roach has moved from a gait with static stability to others with only dynamic stability. When "running" on 4 and 2 legs, its body doesn't actually leave the ground on every step like a human or dog [although it sometimes may], rather it bounces up and down like a pogo stick [Full91].

<| Robot Design

In review, the following principles were discussed here: in dynamic objects, energy can be stored and recovered using both potential [gravitational] and kinetic means, as well as in elastic devices. moving bodies naturally store energy kinetically, but if they are also rising and falling, then potential energy stores are also available. living creatures also store energy in their musculoskeletal structures - by storing energy in their muscles and tendons during ground impact, and releasing it back during subsequent propulsion. clever animals can absorb energy from the substrates in which they locomote - by wind, water, springy ground; this is generally true, but also specifically true for underwater movers, like crabs. animals have to compensate for external forces, such as winds and water currents, and for some, like sprawling-postured arthropods, their leg design provides automatic lateral self-stabilization. self-stabilization can also lead to poor voluntary responsiveness [heavy-weight inertia being one example]; so there is a tradeoff between maneuverability and passive stability - eg, elephants cannot spin on a dime. Robots can prosper from the aspects of animal dynamics, just described, in several ways: robot bodies can be designed to take advantage of potential-kinetic energy transformations, and especially forward inertia. robot legs can be designed to absorb, store, and then re-release the energy of foot impact. robot legs can be arranged, like in the roach, to take advantage of self-stabilizing forces, which in turn can lessen the complexity of controllers and improve the overall stability of the devices in dynamic situations.

<| Bibliography