Quadrupedalism is a form of terrestrial locomotion where a tetrapod (as well as mantises, who primarily walk in this way) animal uses all four limbs ( legs) to bear weight, walk, and run. An animal or machine that usually maintains a four-legged posture and moves using all four limbs is said to be a quadruped (from Latin quattuor for "four", and pes, pedis for "foot"). Most quadrupeds are terrestrial vertebrates, including mammals and reptiles, though some are largely aquatic such as turtles, amphibians, and pinnipeds.
Although the words ‘quadruped’ and ‘tetrapod’ are both derived from terms meaning ‘four-footed’, they have distinct meanings. A tetrapod is any member of the taxonomic unit Tetrapoda (which is defined by descent from a specific four-limbed ancestor), whereas a quadruped actually uses four limbs for locomotion. Not all tetrapods are quadrupeds and not all entities that could be described as ‘quadrupedal’ are tetrapods. This last meaning includes certain artificial objects; almost all quadruped organisms are tetrapods (with the exception of some raptorial arthropods adapted for four-footed locomotion, such as the Mantodea).
The distinction between quadrupeds and tetrapods is important in evolutionary biology, particularly in the context of tetrapods whose limbs have adapted to other roles (e.g., hands in the case of humans, wings in the case of birds, and fins in the case of whales). All of these animals are tetrapods, but none is a quadruped. Even snakes, whose limbs have become vestigial or lost entirely, are, nevertheless, tetrapods.
In July 2005, in rural Turkey, scientists discovered five Turkish siblings who had learned to walk naturally on their hands and feet. Unlike chimpanzees, which ambulate on their knuckles, the Ulas Family walked on their palms, allowing them to preserve the dexterity of their fingers.   
Also by NASA JPL, in collaboration with University of California, Santa Barbara Robotics Lab, is RoboSimian, with emphasis on stability and deliberation. It has been demonstrated at the DARPA Robotics Challenge. 
A related concept to quadrupedalism is pronogrady, or having a horizontal posture of the trunk. Although nearly all quadrupedal animals are pronograde, bipedal animals also have that posture, including many living birds and extinct dinosaurs. 
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