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This paper presents a framework for the dynamical formulation and control of humanoid systems. In this framework unactuated virtual joints are used to describe the humanoid's configuration with respect to the inertial frame. The dynamics of the system are then formulated in a general manner that considers arbitrary contact with the environment. A control structure is implemented for both motion and contact forces that accounts for under-actuation due to the virtual joints. A strategy is also implemented to address transitions between different contact states. Simulation results are presented that demonstrate this overall framework for many behaviors such as standing, walking, jumping, and hand manipulation with walking
Park et al. (Mon,) studied this question.