## 2 Motion States of Freestanding Package
Therefore, we obtain  
\[\lim_{t^{+}\to t^{-}}\int_{t^{-}}^{t^{+}}\mathbf{f}(\mathbf{C},\mathbf{K},\tilde{u}_{g})dt =\mathbf{0},\,\lim_{t^{+}\to t^{-}}\int_{t^{-}}^{t^{+}}\mathbf{\lambda}_{N} dt=\mathbf{\Lambda}_{N},\,\lim_{t^{+}\to t^{-}}\int_{t^{-}}^{t^{+}}\mathbf{\lambda}_{T} dt=\mathbf{\Lambda}_{T} \tag{3}\]  
From the integration of Equation (1), we obtain:  
\[\dot{\mathbf{q}}^{+}-\dot{\mathbf{q}}^{-}=\mathbf{M}^{-1}\mathbf{W}_{N}\mathbf{\Lambda}_{N}+\mathbf{M }^{-1}\mathbf{W}_{T}\mathbf{\Lambda}_{T} \tag{4}\]  
Multiplying Equation (4) by the matrices \(\mathbf{W}_{N}{}^{T}\) and \(\mathbf{W}_{T}{}^{T}\), where the superscript T denotes the matrix transpose, letting \(\dot{\mathbf{g}}_{N}=W_{N}^{T}\mathbf{\dot{q}}\) and \(\dot{\mathbf{g}}_{T}=W_{T}^{T}\mathbf{\dot{q}}\) represent the normal and tangential velocities of the package relative to the vehicle base after the impact, respectively, we obtain:  
\[ \begin{array}{l}\dot{\mathbf{S}}_{N}^{+}-\dot{\mathbf{S}}_{N}^{-}=\mathbf{G}_{NN}\mathbf{ \Lambda}_{N}+\mathbf{G}_{NT}\mathbf{\Lambda}_{T}\\ \dot{\mathbf{S}}_{T}^{+}-\dot{\mathbf{g}}_{T}^{-}=\mathbf{G}_{TN}\mathbf{\Lambda}_{N}+\mathbf{G}_{ TT}\mathbf{\Lambda}_{T}\end{array}  \tag{5}\]  
where  
\[ \begin{array}{l}\mathbf{G}_{NN}=\mathbf{W}_{N}^{T}\mathbf{M}^{-1}\mathbf{W}_{N},\ \ \ \ \ \mathbf{G}_{NT}=\mathbf{W}_{N}^{T}\mathbf{M}^{-1}\mathbf{W}_{T}\\ \mathbf{G}_{TN}=\mathbf{W}_{T}^{T}\mathbf{M}^{-1}\mathbf{W}_{N},\ \ \ \ \ \mathbf{G}_{TT}=\mathbf{W}_{T}^{T}\mathbf{M}^{-1}\mathbf{W}_{T}\end{array}  \tag{6}\]  
In package transportation, ensuring transport safety requires measures to prevent cargo sliding in vehicles. Common measures include applying high-friction materials between packages and vehicle surfaces and installing anti-sliding barriers or blocks to restrict displacement. Therefore, assuming the friction coefficient between the package and the vehicle is sufficiently large to prevent sliding motion, we have \(\dot{\mathbf{g}}_{T}=\mathbf{0}\). From Equation (3), we can derive Equation (7):  
\[\mathbf{\Lambda}_{T}=-\mathbf{G}_{TT}^{-1}\mathbf{G}_{TN}\mathbf{\Lambda}_{N} \tag{7}\]  
Here, the Newton contact law is employed to characterize the velocity variation of package impact point: \(\dot{\mathbf{g}}_{N}^{+}=-\varepsilon_{N}\dot{\mathbf{g}}_{N}^{-}\), where \(\dot{\mathbf{g}}_{N}^{+}\) and \(\dot{\mathbf{g}}_{N}^{-}\) represent the normal velocities of the package impact point after and before the impact, respectively, and \(\varepsilon_{N}\) is the restitution coefficient, ranging from 0 (fully plastic impact) to 1 (fully elastic impact). Defining the velocity jump at the package impact point, we obtain:  
\[\mathbf{v}_{N}=\dot{\mathbf{g}}_{N}^{+}+\varepsilon_{N}\dot{\mathbf{g}}_{N}^{-} \tag{8}\]  
Combining Equations (6)-(8), we obtain:  
\[\mathbf{v}_{N}=(\mathbf{G}_{NN}-\mathbf{G}_{NT}\mathbf{G}_{TT}^{-1}\mathbf{G}_{TN})\mathbf{\Lambda}_{N }+(\mathbf{E}+\varepsilon_{N})\dot{\mathbf{g}}_{N}^{-} \tag{9}\]  
\(\mathbf{E}\) is the identity matrix. When package sliding is considered, additional potential motion states emerge during transportation. In Equation (5), \(\dot{\mathbf{g}}_{T}\
eq\mathbf{0}\), and thus, Equation (7) no longer holds. To fully characterize the system dynamics, the relationship between tangential and normal motion must be derived. This analysis will enable the determination of possible complex motion states, presenting an interesting direction for future research. As illustrated in Figure 1, the package and vehicle collide at either point 1 or point 2. At the rotating pivot point of the package, because the package maintains contact with the vehicle, therefore, the velocity jump at the pivot point \(v_{Ni}=0\), the normal impulse force at this point \(\Lambda_{Ni}>0\), and the following condition holds:  
\[ \begin{array}{l}v_{Ni}=\dot{g}_{Ni}^{+}+\varepsilon_{N}\dot{g}_{Ni}^{-}=0\\ \Lambda_{Ni}\geq 0\end{array}