## 5 Dynamic Maneuverability Analysis  
The concept of dynamic maneuverability measurement for AURVs can be defined by adapting [PERSON]'s approach for robotic arms [10]. While the theory of dynamic manipulability measurement of robotics arms is proposed as an index of their ability in manipulating the end effector, the dynamic maneuverability of an underwater vehicle takes into account a quantitative study of the AURV ability to move along its degrees of freedom in the several configurations, by considering the vehicle system dynamics. More in detail, the Equation (25) can be reformulated as follows:  
\[\dot{\mathbf{v}}+\dot{\mathbf{v}}_{\Gamma}(\mathbf{v}_{r},\mathbf{q},\mathbf{\eta}) = J(\mathbf{q})\dot{\mathbf{n}}^{2} \tag{29}\] \[J(\mathbf{q}) = M^{-1}(\mathbf{q})T(\mathbf{q})\mathbf{sgn}(\hat{\mathbf{n}})\] (30) \[\dot{\mathbf{v}}_{\Gamma}(\mathbf{v}_{r},\mathbf{q},\mathbf{\eta}) = M^{-1}\Gamma(\mathbf{v}_{r},\mathbf{q},\mathbf{\eta}) \tag{31}\]  
where \(J(\mathbf{q})\in\mathbb{R}^{6\times m}\) is the linear map between the _power-normalized_ rotational speed vector \(\hat{\mathbf{n}}^{2}\) and the vehicle body-fixed acceleration, including the contribution \(\dot{\mathbf{v}}_{\Gamma}(\mathbf{v}_{r},\mathbf{q},\mathbf{\eta})\) from the nonlinear terms \(\Gamma(\mathbf{v}_{r},\mathbf{q},\mathbf{\eta})\) in Equation (25). According to [PERSON]'s theory [10], the concept of maneuverability can be obtained by observing how the Jacobian matrix \(J(\mathbf{q})\) is deformed by the unit sphere \(\mathbb{S}(\cdot)\in\mathbb{R}^{m}\) of the power-normalized rotational speeds, such as  
\[\mathbb{S}(\hat{\mathbf{n}})=\{\hat{\mathbf{n}}:\|\hat{\mathbf{n}}\|=1\}. \tag{32}\]  
It is worth noticing how the operator \((\cdot)^{2}\), defined in Equation (26), transforms the unit sphere \(\mathbb{S}(\cdot)\) in a biquadratic surface \(\mathbb{Q}(\hat{\mathbf{n}})=\{\hat{\mathbf{n}}:\|\hat{\mathbf{n}}^{2}\|=1\}\). In Figure 4, the particular case of two-dimensional biquadratic surface \(\mathbb{Q}(\hat{\mathbf{n}})\) can be observed; from a practical point of view, this situation corresponds to afford the problem of estimating the maximum values of the AURV body accelerations \(\dot{\mathbf{v}}\) given the quasi-maximum value of the thruster rotational speeds. For the sake of completeness, the term \"quasimaximum\" is used in this work to outline the fact the system inputs will never reach their maximum value at the same time, as shown in Figure 4.  
Turning to a geometric interpretation of \(\mathbb{Q}(\hat{\mathbf{n}})\), the set of all propeller rotational speeds, at the input to the dynamic system defined in Equation (31), can be formulated as  
\[\left(\hat{\mathbf{n}}^{2}\right)^{T}\hat{\mathbf{n}}^{2}=1 \tag{33}\]  
where \(\hat{\mathbf{n}}^{2}\) is defined as  
\[\hat{\mathbf{n}}^{2}=J(\mathbf{q})^{\dagger}(\dot{\mathbf{v}}+\dot{\mathbf{v}}_{\Gamma}(\mathbf{v }_{r},\mathbf{q},\mathbf{\eta})) \tag{34}\]  
Figure 4: The two\(-\)dimensional biquadratic surface \(\mathbb{Q}(\hat{\mathbf{n}})\). which leads to  
\[(\dot{\
u}+\dot{\
u}_{\Gamma}(\
u_{r},\mathbf{q},\mathbf{\eta}))^{T}((J(\mathbf{q})^{\dagger}) ^{T}J(\mathbf{q})^{\dagger})(\dot{\
u}+\dot{\
u}_{\Gamma}(\
u_{r},\mathbf{q},\mathbf{\eta}))=1 \tag{35}\]  
where the operator \((\cdot)^{\dagger}\) refers to the Moore-Penrose pseudoinverse. The Equation (35) represents the dynamic maneuverability ellipsoid (DME) of the dynamic system described in Equation (31), defined as the set of the possible body-fixed accelerations \(\dot{\
u}\) given the biquadratic surface \(\mathbb{Q}(\dot{\mathbf{n}})\) as input, and the acceleration \(\dot{\
u}_{\Gamma}(\
u_{r},\mathbf{q},\mathbf{\eta})\) provided by the nonlinear forces. As a matter of fact, the nonlinear accelerations \(\dot{\
u}_{\Gamma}(\
u_{r},\mathbf{q},\mathbf{\eta})\) result in a translation of the DME center, while it can be observed how the deformation (i.e., the length of the semiaxis) of the DME is related to the Jacobian \(J(\mathbf{q})=M^{-1}(\mathbf{q})T(\mathbf{q})\mathbf{sgn}(\dot{\mathbf{n}})\).