## 3 Control of Underactuated Systems with Variant Constraints
Owing to the assumptions (A1) and (A2), without loss of generality, both the passive subsystems and actuated subsystems of Equations (21) and (22) can be changed to a linear system with proper dimension by certain globally diffeomorphic coordinate transformations and input changes [26], and the corresponding equivalent system can be expressed as the following switched linear systems [22]  
\[\dot{\mathbf{x}}_{\sigma}=\mathbf{A}_{\sigma}\mathbf{x}_{\sigma}+\mathbf{B}_{ \sigma}\mathbf{v}_{\sigma}, \tag{30}\]  
where \(\sigma(t)\in H\), \(H=\{1,2,\ldots,N_{m}\}\) is a piecewise constant function of time, called a switching signal, and \(N_{m}\) is the total number of the continuous time subsystems of the hybrid dynamical system (21), (22), (23) or (24). For the passive subsystems of the hybrid dynamical system (21), (22), (23) or (24), the state variables in Equation (30) are given by \(\mathbf{x}_{\sigma}=\text{col}(\mathbf{q}_{x},\dot{\mathbf{q}}_{x})\) and the equivalent inputs are \(\mathbf{v}_{\sigma}=\ddot{\mathbf{q}}_{x}\), while, for the actuated subsystems of the hybrid dynamical system (21), (22), (23) or (24), the state variables and the inputs are given by \(\mathbf{x}_{\sigma}=\text{col}(\mathbf{q}_{s},\dot{\mathbf{q}}_{s})\) and \(\mathbf{v}_{\sigma}=\ddot{\mathbf{q}}_{s}\), respectively. For \(\forall\sigma(t)\in H\), the two matrices \(\mathbf{A}_{\sigma}\) and \(\mathbf{B}_{\sigma}\) are constant and the pair \((\mathbf{A}_{\sigma},\mathbf{B}_{\sigma})\) is controllable. We consider the trajectory tracking problems for the hybrid system (21), (22), (23) or (24). Suppose \(\mathbf{x}^{d}(t)=\text{col}(\mathbf{q}^{d}_{x},\mathbf{q}^{d}_{s})\) is a given target trajectory of the hybrid system (21), (22), (23) or (24) on \(t\in[0,a]\), where \(a>0\) is a constant and \(\text{col}(,)\) represents a column vector. Define a set \(S=\{1,2,\ldots,N_{\sigma}\}\), where \(N_{\sigma}\) denotes the number of the discrete jumps of systems (21), (22), (23) or (24) along the trajectory \(\mathbf{x}^{d}(t)\), and define \(t_{i}=t_{i}^{+}=t_{i}^{-}\) (\(i\in S\)) to be the instant of the state jumps,and \(\Delta t_{vc}=t_{i}^{+}-t_{i}^{-}=0\) due to the assumption (A3). Let \(\Delta t_{i}=\left[ \begin{array}{cc}t_{i-1}^{+}&t_{i}^{-}\end{array} \right]\) with all \(\Delta t_{i}>\tau^{*}>0\), where \(\tau^{*}\) is a constant, and \(a=\sum\limits_{i=1}^{N_{\sigma}}\Delta t_{i}\); then, based on the basic assumptions, a switching controller for the hybrid systems (21), (22), (23) or (24) can be presented as follows. If the hybrid systems (21), (22), (23) or (24) satisfy the basic assumptions, and, for a given state trajectory \(\mathbf{x}^{d}(t)=\mathrm{col}(\mathbf{x}_{\mathbf{x}}^{d},\mathbf{x}_{ \mathbf{s}}^{d})=\mathrm{col}(\mathbf{q}_{\mathbf{x}}^{d},\mathbf{q}_{ \mathbf{s}}^{d})\) on \(t\in[0,a]\), there exists a set of piecewise controllers \(\mathbf{v}_{\sigma(\Delta t_{i})}(t)\), \(\sigma(\Delta t_{i\in S})\in H\) so that  
1. on the time intervals \(t\in\Delta t_{i\in\Omega\subset S}\) and \(\Omega\
eq\varnothing\), the errors of the closed-loop subsystem \(\mathbf{e}_{s}=\mathbf{x}_{s}(t)-\mathbf{x}_{s}^{d}(t)\) controlled by the inputs \(\mathbf{v}_{\sigma(\Delta t_{i})}(t)\), \(t\in\Delta t_{i\in\Omega\subset S}\) are globally exponentially stable and meanwhile the errors of the open-loop subsystem \(\mathbf{e}_{x}=\mathbf{x}_{x}(t)-\mathbf{x}_{x}^{d}(t)\) do not increase, and
2. on the intervals \(t\in\Delta t_{i\in S-\Omega}\) and \(S-\Omega\
eq\varnothing\), the errors of the closed-loop subsystem \(\mathbf{e}_{x}=\mathbf{x}_{x}(t)-\mathbf{x}_{x}^{d}(t)\) controlled by the inputs \(\mathbf{v}_{\sigma(\Delta t_{i})}(t)\), \(t\in\Delta t_{i\in S-\Omega}\) are globally exponentially stable and meanwhile the errors of the open-loop subsystem \(\mathbf{e}_{s}=\mathbf{x}_{s}(t)-\mathbf{x}_{s}^{d}(t)\) do not increase. In addition,
3. if the hybrid systems (21), (22), (23) or (24) are slow switching systems, that is, the constant \(\tau^{*}\) is sufficiently large.