## 2 Hybrid Dynamics of Underactuated Systems with Variant Constraints
In addition, an assumption \(\mathbf{B}=(0,\mathbf{I}_{l})\in\Re^{n\times s}\) is also considered in Equation (18) without losing any generality. For the underactuated system (18) with variant constraints Equation (2), we propose the following corollary based on Theorem 1. **Corollary 1**. : _(Impulse calculation for underactuated systems with variant constraints)_  
_If all the external variables of the underactuated mechanical system (18) are passive, all of the shape variables are actuated, and the constraint variations are instantaneously completed, i.e., \(\Delta t_{vc}\approx 0\), then, for the underactuated system (18), the impulse defined by Equation (4) can be calculated as_  
\[\mathbf{\tilde{\lambda}}=-\Big{[}\mathbf{W}^{\mathrm{T}}(\mathbf{q})\mathbf{M}^{- 1}\mathbf{W}(\mathbf{q})\Big{]}^{-1}\mathbf{W}^{\mathrm{T}}(\mathbf{q})\dot{ \mathbf{q}}(t_{vc}^{-}) \tag{19}\]  
_for adding constraints, and_  
\[\mathbf{\tilde{\lambda}}=\Big{[}\mathbf{W}^{\mathrm{T}}(\mathbf{q})\mathbf{M}^{- 1}\mathbf{W}(\mathbf{q})\Big{]}^{-1}\mathbf{W}^{\mathrm{T}}(\mathbf{q})\dot{ \mathbf{q}}(t_{vc}^{+}) \tag{20}\]  
_for reducing constraints._  
**Remark 1**. : _The impulse \(\mathbf{\tilde{\lambda}}\) defined in Equation (4) is an important parameter for analyzing the dynamical behaviors or designing a closed-loop controller for a variant constraint system, which is generally a hybrid dynamical system. Since the impulse \(\mathbf{\tilde{\lambda}}\) could not be directly calculated by its definition Equation (4) due to the uncertainties of the time interval \(\Delta t_{vc}=t_{vc}^{+}-t_{vc}^{-}\) in practice, Equations (19) and (20) provide a feasible way to explicitly calculate the impulse \(\mathbf{\tilde{\lambda}}\) caused by the constraint variations for underactuated systems, or Equations (7) and (8) for general dynamical systems._  
By combining Equations (2), (13), (18) and (19)/(20), the underactuated systems with instantaneously variant constraints can be presented as follows:  
Least constraint mode:  
\[\left\{ \begin{array}{l}\frac{\mathrm{d}}{\mathrm{d}t}\frac{\partial T}{ \partial\mathbf{q}_{x}}-\frac{\partial V}{\partial\mathbf{q}_{x}}=0,\\ \frac{\mathrm{d}}{\mathrm{d}t}\frac{\partial T}{\partial\mathbf{q}_{y}}-\frac{ \partial L}{\partial\mathbf{q}_{y}}=\mathbf{u},\end{array} \right. \tag{21}\]  
Full constraint mode:  
\[\left\{ \begin{array}{l}\frac{\mathrm{d}}{\mathrm{d}t}\frac{\partial T}{ \partial\mathbf{q}_{x}}+\frac{\partial V}{\partial\mathbf{q}_{x}}=\mathbf{W} _{x}(\mathbf{q})\lambda,\\ \frac{\mathrm{d}}{\mathrm{d}t}\frac{\partial T}{\partial\mathbf{q}_{x}}-\frac{ \partial L}{\partial\mathbf{q}_{y}}=\mathbf{W}_{s}(\mathbf{q})\lambda+ \mathbf{u},\\ \mathbf{W}^{\mathrm{T}}(\mathbf{q})\dot{\mathbf{q}}=0,\end{array} \right. \tag{22}\]  
Discrete jumps:  
\[\left\{ \begin{array}{l}\dot{\mathbf{q}}(t_{vc}^{+})=\dot{\mathbf{q}}(t_{vc}^ {-})+\mathbf{M}^{-1}\mathbf{W}(\mathbf{q})\mathbf{\tilde{\lambda}},\\ \mathbf{\tilde{\lambda}}=-\big{(}\mathbf{W}^{\mathrm{T}}(\mathbf{q})\mathbf{M}^{- 1}\mathbf{W}(\mathbf{q})\big{)}^{-1}\mathbf{W}^{\mathrm{T}}(\mathbf{q})\dot{ \mathbf{q}}(t_{vc}^{-}),\end{array} \right. \tag{23}\]  
or  
\[\left\{ \begin{array}{l}\dot{\mathbf{q}}(t_{vc}^{+})=\dot{\mathbf{q}}(t_{vc}^ {-})+\mathbf{M}^{-1}\mathbf{W}(\mathbf{q})\mathbf{\tilde{\lambda}},\\ \mathbf{\tilde{\lambda}}=\big{(}\mathbf{W}^{\mathrm{T}}(\mathbf{q})\mathbf{M}^{- 1}\mathbf{W}(\mathbf{q})\big{)}^{-1}\mathbf{W}^{\mathrm{T}}(\mathbf{q})\dot{ \mathbf{q}}(t_{vc}^{+}).\end{array} \right. \tag{24}\]  
The hybrid dynamical systems (21), (22), (23) or (24) represent broad classes of underactuated systems in real world applications [1; 3; 4; 5; 6; 7; 8; 9], and can be used in dynamics analysis, motion planning or designing the controllers. For many applications, energy efficiency is an important measurement index, and it is necessary to analyze energy changes caused by the constraint variations.