It's amazing how complex walking motion is! I may be wrong about this (I certainly can't prove that I'm right), but it seems to me that a quadruped needs at least 12 motorized joints to achieve full animal-like walking capabilities. Now, I know that many people have found great ways to achieve quadrupedal walking motion with much fewer motors, but I think these robots have limitations that prevent true animal-like motion. One possible setup of motorized joints in a quadruped could be eight "shoulder" joints and four "elbow" joints, as in the following figure:The type 1 shoulder joints and elbow joints are, of course, for lifting, extending, and lowering the feet during steps. The need for the type 2 shoulder joints is a bit more subtle. Take a look at an above view of a quadruped, showing its center of gravity (COG):
The square formed by the feet represents the "support zone" of the robot. As long as the COG is within this zone, the robot should stay balanced. The closer the COG is to the center of the zone, the more stable the robot will be. Therefore, the current pose is a very stable one. Now, if we lift the front-right foot up to start taking a step, look at the position of the COG relative to the new support zone (the triangle) formed by the remaining three feet:
Now the COG is right on the edge of the support zone. This means the robot is just on the verge of tipping over, and is very unstable. In order to fix this problem, we need a way to shift the COG inside the triangular zone. One way of doing this is to use the type 2 shoulder joints to side-shift the robot away from the foot being lifted. Although there are other ways to shift the center of gravity that use less than four motors, the type 2 shoulder joints would also enable the robot to realistically turn and do other animal-like motions.
One amazing example of an animal-like quadruped is the "Big Dog" by Boston Dynamics. You can see a video of it here.