## 2 Applied Theory
### [PERSON] Dynamic Modeling Methods  
An accurate and reasonable model is the basis for analyzing the dynamic response of multi-joint morphing UAVs. The assumptions of the single rigid-body model are no longer applicable to large-scale morphing UAVs. Therefore, a new dynamic model must be developed. [PERSON]'s methods choose a series of independent generalized velocities to describe the motion of the system. It is no longer necessary to calculate the energy function and related time derivatives to obtain the differential equations of the system without an ideal constraint reaction force [37]. Therefore, it is more suitable for targeted UAV. The specific modeling methods are as follows:  
\(f\) independent variables are selected as the system's generalized velocities, which are expressed as \(u_{k}\) (\(k=1,\,2,\,\ldots,f\)). The translational velocity \(\mathbf{v}_{ci}\) and angular velocity \(\mathbf{\omega}_{i}\) of the \(i\)-th rigid body, relative to the centroid of the inertial reference system, can be separately expressed as a linear combination of the generalized velocities, as follows:  
\[\mathbf{v}_{ci}=  
velocity of the centroid of the \(i\)-th rigid body. For a multi-body system with \(N\) rigid bodies and \(f\) degrees of freedom, [PERSON]'s equation can be expressed as follows:  
\[\mathbf{F}_{k}+\mathbf{F}_{k}^{*}=0,\ \ \ k=1,2,\cdots,f \tag{2}\]  
where \(\mathbf{F}_{k}\) and \(\mathbf{F_{k}}^{*}\) are the generalized active force and generalized inertial force corresponding to \(\mathbf{u}_{k}\), respectively.  
\[\mathbf{F}_{k}=\sum_{i=1}^{N}\Bigl{[}\mathbf{F}_{i}\cdot\mathbf{v}_{ci}^{(k)}+\mathbf{M}_{i} \cdot\mathbf{\omega}_{i}^{(k)}\Bigr{]},\ \mathbf{F}_{k}^{*}=\sum_{i=1}^{N}\Bigl{[}\mathbf{F}_{i}^{*} \cdot\mathbf{v}_{ci}^{(k)}+\mathbf{M}_{i}^{*}\cdot\mathbf{\omega}_{i}^{(k)}\Bigr{]} \tag{3}\]  
where \(\mathbf{F}_{i}\) and \(\mathbf{F_{i}}^{*}\) are the principal vectors of the active force and inertial force acting on the \(i\)-th rigid body, respectively; and \(\mathbf{M}_{i}\) and \(\mathbf{M}_{i}^{*}\) are the principal moments of the active force and inertial force, relative to the centroid of the \(i\)-th rigid body, respectively. They can be expressed as follows:  
\[\mathbf{F}_{i}^{*}=-m_{i}\mathbf{a}_{ci},\ \mathbf{M}_{i}^{*}=-j_{i}\cdot\dot{\mathbf{\omega}}_{i }-\mathbf{\omega}_{i}\times\bigl{(}j_{i}\cdot\mathbf{\omega}_{i}\bigr{)} \tag{4}\]  
where \(j_{i}\) is the inertial tensor and \(\mathbf{a}_{ci}\) is the acceleration of the centroid of the \(i\)-th rigid body. \(\omega_{i}\) is the angular velocity of the \(i\)-th rigid body. If three perpendicular unit vectors, \(\mathbf{c}_{1}\), \(\mathbf{c}_{2}\), and \(\mathbf{c}_{3}\), form a right-handed system and are parallel to the central principal axis of the reference frame (the right-handed system is not required to be fixed), they can be defined as follows:  
\[\left.\begin{array}{l}\dot{\mathbf{\omega}}_{j}=\dot{\mathbf{\omega}}\cdot\mathbf{c}_{ j}\\ \mathbf{\omega}_{j}=\mathbf{\omega}\cdot\mathbf{c}_{j}\\ J_{j}=\mathbf{c}_{j}\cdot\mathbf{c}_{j}\end{array}\right\},\ (j=1,2,3) \tag{5}\]  
Then, the second equation of Equation (4) can be rewritten as follows:  
\[\mathbf{T}_{i}^{*}=\begin{array}{l}-\bigl{[}\dot{\omega}_{i1}\cdot J_{i1}- \omega_{2}\omega_{i3}\bigl{(}j_{l2}-J_{3}\bigr{)}\bigr{]}\mathbf{c}_{i1}\\ -\bigl{[}\dot{\omega}_{i2}\cdot J_{l2}-\omega_{i2}\omega_{i1}\bigl{(}j_{l3}- J_{i1}\bigr{)}\bigr{]}\mathbf{c}_{i2}\\ -\bigl{[}\dot{\omega}_{i3}\cdot J_{i3}-\omega_{i1}\omega_{i2}\bigl{(}j_{l1}-J_{ l2}\bigr{)}\bigr{]}\mathbf{c}_{i3}\end{array} \tag{6}\]