## 2 Hybrid Dynamics of Underactuated Systems with Variant Constraints
The changes of the kinetic energy of the underactuated system (21), (22), (23) or (24) due to the state jumps can be written as  
\[\Delta T(\mathbf{q},\dot{\mathbf{q}})=\frac{1}{2}\left[\dot{\mathbf{q}}^{ \mathrm{T}}\mathbf{M}(\mathbf{q}_{s})\dot{\mathbf{q}}\right]_{t_{vc}^{-}}^{t_{vc }^{+}}. \tag{25}\]By substituting Equation (13) into Equation (25) and applying Equations (14) and (15), it is easy to show that  
\[\Delta T(\mathbf{q},\dot{\mathbf{q}})=\frac{1}{2}\big{(}\mathbf{W}(\mathbf{q}) \tilde{\mathbf{\lambda}}\big{)}^{\mathrm{T}}\mathbf{M}(\mathbf{q}_{s})\mathbf{ M}^{-1}\left[\dot{\mathbf{q}}(t_{\mathrm{rc}}^{+})+\dot{\mathbf{q}}(t_{ \mathrm{rc}}^{-})\right]. \tag{26}\]  
From Equation (26), a useful result for the problems of motion planning and controller design of the hybrid dynamical system (21), (22), (23) or (24) can be presented as follows. **Theorem 2**. : _(A necessary condition for hybrid systems (21), (22), (23) or (24) with energy efficient motions)_  
_For the hybrid dynamical system (21), (22), (23) or (24), if there is a state trajectory \(\mathbf{x}(t)=\left[ \begin{array}{cc}\mathbf{q}&\dot{\mathbf{q}}\\ \end{array} \right]^{\mathrm{T}}(t)\) on \(t\in[0,\infty)\) and \(\lim\limits_{t\to\infty}\dot{\mathbf{q}}(t)\
eq 0\) so that \(\mathbf{u}(t)=0\), that is, if the trajectory \(\mathbf{x}(t)\) is a nontrivial periodical trajectory governed by the passive dynamics of systems (21), (22), (23) or (24), then for all \(t\in\left[ \begin{array}{cc}t_{\mathrm{rc}}^{-}&t_{\mathrm{rc}}^{+}\\ \end{array} \right]\) the impulse caused by the constraint variations satisfies_  
\[\tilde{\mathbf{\lambda}}(t_{\mathrm{rc}})=0. \tag{27}\]  
Proof. : Suppose \(\tilde{\mathbf{\lambda}}(t_{\mathrm{rc}})\
eq 0\) for all \(t\in\left[ \begin{array}{cc}t_{\mathrm{rc}}^{-}&t_{\mathrm{rc}}^{+}\\ \end{array} \right]\), referring to Equation (26), we can conclude that \(\Delta T(\mathbf{q},\dot{\mathbf{q}})\
eq 0\) since the inertia matrix \(\mathbf{M}(\mathbf{q}_{s})\) is a positive definite matrix and \(\mathbf{W}(\mathbf{q})\) is not equal to \(0\) for a variant constraint system. Accordingly, if the kinetic energy \(\Delta T(\mathbf{q},\dot{\mathbf{q}})\
eq 0\) on \(\forall t\in\left[ \begin{array}{cc}t_{\mathrm{rc}}^{-}&t_{\mathrm{rc}}^{+} \\ \end{array} \right]\), then there must be \(\Delta T(\mathbf{q},\dot{\mathbf{q}})<0\) since the trajectory \(\mathbf{x}(t)\) is governed by the passive dynamics of systems (21), (22), (23) or (24). At the moment, it is necessary that \(\lim\limits_{t\to\infty}T(\mathbf{q},\dot{\mathbf{q}})=\frac{1}{2}\lim \limits_{t\to\infty}\left[\dot{\mathbf{q}}^{\mathrm{T}}\mathbf{M}(\mathbf{q}_{ s})\dot{\mathbf{q}}\right]=0\). Consequently, because of the inertia matrix \(\mathbf{M}(\mathbf{q}_{s})\
eq 0\) for all \(t\in[0,\infty)\), we can conclude that \(\lim\limits_{t\to\infty}\dot{\mathbf{q}}(t)=0\) if \(\tilde{\mathbf{\lambda}}(t_{\mathrm{rc}})\
eq 0\) for all \(t\in\left[ \begin{array}{cc}t_{\mathrm{rc}}^{-}&t_{\mathrm{rc}}^{+}\\ \end{array} \right]\). This contradicts the given condition \(\lim\limits_{t\to\infty}\dot{\mathbf{q}}(t)\
eq 0\). Thus, the proof is completed. **Remark 2**. : _It is worth pointing out that the hybrid dynamical system (21), (22), (23) or (24) is obtained based on the assumption \(\Delta t_{\mathrm{rc}}\approx 0\). Otherwise, the dynamical systems should be expressed in different forms. However, modeling the dynamics of the slow varying constraint systems is still an open problem so far._  
**Remark 3**. : _On a passive dynamics trajectory \(\mathbf{x}(t)\), there is no energy consumption because of \(\mathbf{u}(t)=0\). Thus, a nontrivial periodical trajectory \(\mathbf{x}(t)\) governed by the passive dynamics of a hybrid dynamical system (21), (22), (23) or (24) is useful for all mechanical systems if the time-varying trajectory \(\mathbf{x}(t)\) is physically meaningful._