It's been a few years since studying in this area, but one approach we used to use for modelling and control of manipulators (mostly industrial robotic arms) was Screw Theory.

Here's a recent paper using one of these approaches: http://www.intechopen.com/books/international_journal_of_adv...

One approach that could be used for a system like Big Dog would be a feed-forward control loop with kinematic/dynamic modelling of the robot. These approaches use knowledge of the system dynamics to predict the output based on changing inputs or disturbances.