Quadrupeds walk with a duty factor above 1/2 (each foot is on the ground for more than half the stride cycle), but with a range of phases between back and front feet. THIS PAPER describes how the phasing observed, and how it relates to duty factor, matches mechanical work minimisation. It provides an account for the different phasing of ‘normal’ quadrupedal walking, with a duty factor around 0.6 and a phase at or below 25% (HL-FL—HR-FR–HL); see this camel:
An alternative, potentially complementary but based on very different reasoning, account for the near-trotting phasing of very slow (‘grazing’) gaits is described in THIS PAPER. The geometry here is that of a toppling table: once a front leg is picked up, the back (the table top) topples diagonally on to the new stance leg, thereby lifting the diagonally opposite hip; the raised hip lifts the hindleg off the ground, and it swings forwards immediately.
This idea is also quite distinct from those of stability and maintaining the centre of mass over some polygon of support. Indeed, grazing animals often lift both diagonal feet off the ground at the same time, even when they are moving very, very slowly.
In THIS PAPER, Ben Smith describes a ‘minimalist analague robot’ with table-like characteristics that naturally ‘discovers’ the grazing gait, showing how animal-like gaits can be achieved with totally decentralised control. Each robot leg has a force sensor and motor, and the rule ‘swing backwards slowly when loaded’ and ‘swing forwards quickly when unloaded’.
This is effectively a form of ‘reflex-driven’ control – which ties in with the ‘legs as linkages‘ and ‘human legs and feet‘ pages.