## 5 Dynamic Maneuverability Analysis
In light of these considerations, the volume of the DME only depends on the chosen propulsion system, on the physical properties of the AURV, and on the thruster configuration on the actual configuration. A more exhaustive explanation can be provided by considering the singular value decomposition (SVD) of \(J(\mathbf{q})\)[19],  
\[J = U\Sigma V^{T}=U\bigg{[} \begin{array}{cc}\mathrm{diag}_{r\times r }(\sigma_{1},\ldots,\sigma_{r})&0_{(r)\times(m-r)}\\ 0_{(6-r)\times r}&0_{(6-r)\times(m-r)}\end{array} \bigg{]}V^{T} \tag{36}\] \[J^{+} = V\Sigma^{+}U^{T}=V\bigg{[} \begin{array}{cc}\mathrm{diag}_{r \times r}(1/\sigma_{1},\ldots,1/\sigma_{r})&0_{(r)\times(6-r)}\\ 0_{(n-r)\times r}&0_{(m-r)\times(6-r)}\end{array} \bigg{]}U^{T} \tag{37}\]  
where \(\Sigma\in R^{6\times m}\) contains the \(r\) nonzero singular values \(\sigma_{i}\) of \(J(\mathbf{q})\) in decreasing order, \(U\in\mathbb{R}^{6\times 6}\) is an orthogonal matrix whose columns are the left singular vectors of \(J(\mathbf{q})\) that form an orthonormal base of the output space, and \(V\in\mathbb{R}^{m\times m}\) is an orthogonal matrix whose columns are the right singular vectors of \(J(\mathbf{q})\) that form an orthonormal base of the input space. Furthermore, considering that  
\[\Sigma^{+}\in\mathbb{R}^{m\times 6}:\Sigma^{+} = \bigg{[} \begin{array}{cc}\mathrm{diag}_{r\times r}(1/\sigma_{ 1},\ldots,1/\sigma_{r})&0_{(r)\times(6-r)}\\ 0_{(n-r)\times r}&0_{(m-r)\times(6-r)}\end{array} \bigg{]} \tag{38}\] \[\Sigma^{+T}\in\mathbb{R}^{6\times m}:\Sigma^{+T} = \bigg{[} \begin{array}{cc}\mathrm{diag}_{r\times r}(1/\sigma_{ 1},\ldots,1/\sigma_{r})\vdots&0_{(r)\times(m-r)}\\ 0_{(6-r)\times r}&0_{(6-r)\times(m-r)}\end{array} \bigg{]}\] (39) \[\Big{(}\Sigma^{+T}\Sigma^{+}\Big{)} = \bigg{[} \begin{array}{cc}\mathrm{diag}_{r\times}(1/\sigma_{1}^ {2},\ldots,1/\sigma_{r}^{2})&0_{(r)\times(6-r)}\\ 0_{(6-r)\times r}&0_{(6-r)\times(6-r)}\end{array} \bigg{]}\in\mathbb{R}^{6 \times 6}\quad, \tag{40}\]  
and inserting Equations (37), (36), and (40) in Equation (35), it results in the formulation of the DME expressed in the base of the left singular vectors of \(\dot{\
u}+\dot{\
u}_{\Gamma}(\
u_{r},\mathbf{q},\mathbf{\eta})\), as follows  
\[\mathbf{w}^{T}\Big{(}\Sigma^{+T}\Sigma^{+}\Big{)}\mathbf{w} = 1 \tag{41}\] \[\mathbf{w} = U^{T}(\dot{\
u}+\dot{\
u}_{\Gamma}(\mathbf{v}_{r},\mathbf{q},\mathbf{\eta})),\quad\mathbf{w}\in\mathbb{R}^{6}\quad. \tag{42}\]  
Equation (41) describes the relationship among the singular values \(\sigma_{i}\) of \(J(\mathbf{q})\) and the DME. In greater detail, it can be observed that the singular values \(\sigma_{i}\) represent the length of the DME semiaxis, expressed as in Equation (41), whereas their direction in the body-fixed frame can be obtained from the \(i\)th column of the matrix \(U=[\cdots\mathbf{U}_{i}\cdots]\). Finally, from a practical consideration, the DME expressed in Equation (35) is studied in a decoupled manner from the linear \(\dot{\
u}_{1}\) and angular \(\dot{\
u}_{2}\) accelerations point of view. Thus, by defining  
\[J(\mathbf{q})=\left[ \begin{array}{c}J_{1}(\mathbf{q})\\ J_{2}(\mathbf{q})\end{array} \right]\quad, \tag{43}\]the linear DME and the angular DME are evaluated, such as  
\[(\dot{\mathbf{v}}_{1}+\dot{\mathbf{v}}_{\rm T1}(\mathbf{v}_{r},\mathbf{q},\mathbf{\eta}))^ {T}((f^{\dagger}_{1}(\mathbf{q}))^{T}f^{\dagger}_{1}(\mathbf{q}))(\dot{\mathbf{v}}_{1}+\dot{ \mathbf{v}}_{\rm T1}(\mathbf{v}_{r},\mathbf{q},\mathbf{\eta})) = 1 \tag{44}\] \[(\dot{\mathbf{v}}_{2}+\dot{\mathbf{v}}_{\rm T2}(\mathbf{v}_{r},\mathbf{q},\mathbf{\eta})) ^{T}((f^{\dagger}_{2}(\mathbf{q}))^{T}f^{\dagger}_{2}(\mathbf{q}))(\dot{\mathbf{v}}_{2}+ \dot{\mathbf{v}}_{\rm T2}(\mathbf{v}_{r},\mathbf{q},\mathbf{\eta})) = 1\quad. \tag{45}\]  
For the sake of completeness, the well-known formulations of Equations (41) and (37) can be extended for both the linear and angular ellipsoids defined in Equations (44) and (45).