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.
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.
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.