Embedding the generalized Acrobot into the n-link with an unactuated cyclic variable and its application to walking design

The Acrobot is the well-known and widely studied underactuated mechanical system having two links and one actuated joint between them. It may be also viewed as the simplest possible walking like mechanism without knees and the ankle-joint actuation, alternatively also referred to as the underactuated Compass gait walker. To extend techniques used to control the Acrobot to a more general underactuated n-link having an unactuated cyclic variable, this paper defines the so-called generalized Acrobot. Further, it is shown that for every set of virtual constraints there exists a generalized Acrobot that is linearly embedded into this n-link. Based on this property and results valid for the Acrobot, walking strategies for the n-link are provided. Important achievement here is that the exponentially stable tracking during the swing phase only is possible, i.e. the stabilizing effect of the impact map is not needed. Computer simulations of the 4-link case are provided.