While common in nature, four-legged robots are comparatively few and far between in the robot world. For stability, it’s hard to beat their six-legged hexapod cousins, and if you’re going to reduce the number of limbs beyond this, one could make an argument for simply jumping to a biped/humanoid configuration. Nonetheless, when they’re implemented (e.g. very expensive bots by Boston Dynamics), they can be truly incredible. Creator Mech-Dickel Robotics has pulled off a cat-sized build called mechDOG that doesn't just look great, but is certainly cheaper than those made by larger robotics firms.
As seen in the video below, the mechDOG is able to traverse one meter in 8.46 seconds, which isn’t as fast as an RC car, but not bad as far as walking robots go. It uses three ST1 servos per leg for 12 total, with a shoulder joint assembly that flexes in/out and front/back, plus a shoulder that moves in a front/back orientation as well.
The quadruped is controlled by a Lynxmotion Smart Servo adapter board coupled to an Arduino Uno. A remote control module provides the user interface (actually two styles in the first and second videos) via what appears to be nRF24L01 transceivers. Servo motions were worked out visually with a 2D representation and simulated on-screen protractor, which seems to be a quite valid though time-consuming method.