Action variables, the components of $ u$, are often referred to as actuators, and a system is called underactuated if the number of actuators is strictly less than the dimension of $ {\cal C}$. In other words, there are less independent action variables than the degrees of freedom of the mechanical system. Underactuated nonlinear systems are typically nonholonomic. Therefore, a substantial amount of nonholonomic system theory and planning for nonholonomic systems involves applications to underactuated systems. As an example of an underactuated system, consider a free-floating spacecraft in $ {\mathbb{R}}^3$ that has three thrusters. The amount of force applied by each thruster can be declared as an action variable; however, the system is underactuated because there are only three actuators, and the dimension of $ {\cal C}$ is six. Other examples appeared Section 13.1.2. If the system is not underactuated, it is called fully actuated, which means that the number of actuators is equal to the dimension of $ {\cal C}$. Kinodynamic planning has mostly addressed fully actuated systems.

Steven M LaValle 2020-08-14