## 3 Control of Underactuated Systems with Variant Constraints  
In this section, we discuss the control issues of the underactuated hybrid dynamical system (21), (22), (23) or (24). Due to the [PERSON]'s theorem [12], which points out that a nonholonomic system can not be stabilized by any smooth time invariant pure state feedback, now it is well known that controller design for an underactuated system with a single mode is not even a simple task. Thereafter, to date, almost all controllers presented in literature for nonholonomic systems adopt time-varying feedback or non-smooth feedback. The underactuated systems (21), (22), (23) or (24) are generally some kinds of nonholonomic systems. This is due to the fact that there are some passive degrees of freedom (DOF) in the systems. Regardless of the control input, we have  
\[\frac{\mathrm{d}}{\mathrm{d}t}\frac{\partial T}{\partial\dot{\mathbf{q}}_{x}}+ \frac{\partial V}{\partial\mathbf{q}_{x}}=0 \tag{28}\]  
\[\frac{\mathrm{d}}{\mathrm{d}t}\frac{\partial T}{\partial\dot{\mathbf{q}}_{x}}+ \frac{\partial V}{\partial\mathbf{q}_{x}}=\mathbf{W}_{x}(\mathbf{q})\mathbf{\lambda} \tag{29}\]from Equation (22), and the differential Equations (28) and (29) are not integrable generally (except for some special circumstances such as the external variants \(\mathbf{q}_{x}\) happen to be cyclic coordinates). For the underactuated system (21), (22), (23) or (24), from the viewpoint of control, Equations (28) and (29) are additional differential constraints with respect to the inputs if all state variables of the underactuated systems (21), (22), (23) or (24) are simultaneously stabilized. However, from the viewpoint of mechanics, an underactuated system with partial passive DOF usually satisfies the Lagrange equations without the need for using any external differential constraints [24]. This is different from a traditional nonholonomic system such as a system with rolling constraints that requires the use of Lagrangian multipliers in establishing the dynamics [11]. We notice that a traditional nonholonomic system commonly involves external differential constraints. However, an underactuated system brings about nonholonomic constraints from its own dynamics, such as the underactuated systems that satisfy the law of angular momentum conservation. Therefore, it is more reasonable that a traditional nonholonomic system is renamed as the exogenous nonholonomic systems while an underactuated nonholonomic system with partial passive DOF is named as the endogenous nonholonomic systems. Before discussing the control issues of the systems (21), (22), (23) or (24), it is necessary to point out that the Lagrange multipliers \(\lambda\) in Equation (22) can commonly be eliminated by properly using the dynamics under the least constraint mode and the constraint Equation (2). Then, the state space dimensions of the system (22) under full constraint mode will reduce. Nevertheless, after eliminating the Lagrange multipliers \(\lambda\), the dynamics of full constraint mode can also be expressed as the form of Equation (21), as it should be, and the dimensions of the state space of the continuous time subsystems in double modes are usually different. Now, taking the control problems of the hybrid systems into consideration, some basic assumptions are given as follows:  
Basic assumptions:  
(A1) The actuated DOF \(s\) of the underactuated system (21), (22), (23) or (24) in continuous-time modes is not less than the passive DOF \(n-s\);  
(A2) All the passive subsystems \((\mathbf{q}_{x},\dot{\mathbf{q}}_{x})\) and the actuated subsystems \((\mathbf{q}_{s},\dot{\mathbf{q}}_{s})\) of the hybrid system (21), (22), (23) or (24) in continuous-time modes are respectively globally controllable;  
(A3) The constraint variations of the hybrid system (21), (22), (23) or (24) are instantaneously completed, i.e., \(\Delta t_{vc}\approx 0\).