Balancing is a low-frequency task
Some works strive to implement closed-loop model predictive control as fast as possible, often with balancing as an application example. Computation time as a metric may drive some discoveries, but I don't think it makes sense to take this particular application example, as balancing is a relatively low-frequency task. Bipeds and quadrupeds can balance themselves as leisurely as 5-15 Hz, at a frequency much lower than the typical 200-1000 Hz of joint control. There is both theoretical and empirical evidence of this in (Villa et al., 2019) for an adult-size humanoid robot balanced by DCM feedback. It establishes that the sampling frequency for the balancing task should be at least:
where is the height of the center of mass, is the DCM feedback gain, and is the acceleration due to gravity. For HRP-4 with the LIPM walking controller, and , so that .
I think the observation by Villa et al. has been undervalued so far, both in research and software engineering. For the latter, it is even good news! It means we can implement more functionality in higher-level languages (🦊).
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.