This is a temporary list of ideas pointers that I should eventually draft into robotics notes.
The many divergent components of motion¶
The 2D capture point for the linear inverted pendulum model has been generalized to a 3D divergent component of motion (DCM) by (Englsberger et al., 2013), and applied successfully to both walking pattern generation and balance control. Many extensions to this DCM have been sought as 3D points, but when we deal with more general models than the linear inverted pendulum, the dimension for divergent components of motion can be higher. For example, for the variable-height inverted pendulum, there is a 4D DCM whose feedback control makes the robot crouch or stretch up when needed.
Divergent components of motion in general are related to the concept of expoential dichotomy (Coppel, 1966). More details on the search for higher-order DCMs in this presentation.