# Control of an Omnidirectional UAV for Transportation and Manipulation Tasks
## 5 Stability Analysis
As for the perturbation term, the following property of the inertia matrix can be exploited [34; 38]:  
**Property 1**. : \(\mathbf{M}\) _is symmetric and positive definite; therefore, if \(\lambda_{min}(\cdot)\) (\(\lambda_{max}(\cdot)\)) denotes the minimum (maximum) eigenvalue, it is_  
\[0<\lambda_{min}(\mathbf{M})\mathbf{I}_{8}\leq\mathbf{M}(\mathbf{q})\leq\lambda_{max}(\mathbf{M})\mathbf{ I}_{8}. \tag{40}\]  
Thus, based on Property 1, the following holds:  
\[\lambda_{max}^{-1}(\mathbf{M})<\left\|\mathbf{M}^{-1}\right\|<\lambda_{min}^{-1}(\mathbf{M }). \tag{41}\]  
Regarding matrix \(\widetilde{\mathbf{M}}\), it is worth considering the explicit expression of its sole non-null block (6), i.e.,  
\[\mathbf{M}_{p\delta}=\big{[}\ -\big{(}m_{1}\mathbf{S}(\mathbf{r}_{1,C_{1}})+m_{2}\mathbf{S}( \mathbf{r}_{2,C_{2}})\big{)}\mathbf{x}_{0}\,,\ \ -m_{2}\mathbf{S}(\mathbf{r}_{2,C_{2}})\mathbf{x}_{1}\ \ \big{]}, \tag{42}\]  
where \(\mathbf{x}_{0}\) and \(\mathbf{x}_{1}\) are unit vectors and all the other elements are constant and depend on the system's geometrical and inertial properties. Thus, its norm is constant  
\[\left\|\widetilde{\mathbf{M}}\right\|=\left\|\mathbf{M}_{p\delta}\right\|=\Lambda. \tag{43}\]  
Finally, the term \(\mathbf{\gamma}\) can be written in the explicit form as  
\[\mathbf{\gamma}=\bar{\mathbf{q}}_{d}+\mathbf{K}_{D}\dot{\mathbf{e}}+\mathbf{K}_{P}\mathbf{e}+\mathbf{K}_{ I}\int\mathbf{e}.\]  
The following assumption on the planned trajectory is made:  
**Assumption 1**. : _The planned acceleration \(\bar{\mathbf{q}}_{d}\) is bounded \(\forall\ t\in\mathcal{T}\), i.e., \(\|\bar{\mathbf{q}}_{d}(t)\|\leq\bar{\mathbf{q}}\)._  
Therefore, the non-null block of the perturbation term can be rewritten as sum of two terms  
\[\widetilde{\gamma}_{3}=-\bigg{(}\mathbf{M}^{-1}\widetilde{\mathbf{M}}\bar{\mathbf{q}}_{d }+\mathbf{M}^{-1}\widetilde{\mathbf{M}}\bigg{(}\mathbf{K}_{D}\dot{\mathbf{e}}+\mathbf{K}_{P}\mathbf{e }+\mathbf{K}_{I}\int\mathbf{e}\bigg{)}\bigg{)}. \tag{44}\]  
The first term, in view of (41), (43) and of Assumption 1, is norm-bounded \(\forall\ t\)  
\[\left\|\mathbf{M}^{-1}\widetilde{\mathbf{M}}\bar{\mathbf{q}}_{d}\right\|\leq\frac{1}{ \lambda_{min}(\mathbf{M})}\Lambda\ddot{\bar{\mathbf{q}}}. \tag{45}\]  
The second term in (44) is a vanishing perturbation (in the sense of [39]), since the following chain of inequalities holds:  
\[\left\|\mathbf{M}^{-1}\widetilde{\mathbf{M}}\bigg{(}\mathbf{K}_{D}\dot{\mathbf{e }}+\mathbf{K}_{P}\mathbf{e}+\mathbf{K}_{I}\int\mathbf{e}\bigg{)}\right\| \leq \frac{1}{\lambda_{min}(\mathbf{M})}\Lambda\left\|\mathbf{K}_{D}\dot{\mathbf{e }}+\mathbf{K}_{P}\mathbf{e}+\mathbf{K}_{I}\int\mathbf{e}\right\| \tag{46}\] \[\leq \frac{1}{\lambda_{min}(\mathbf{M})}\Lambda k_{M}\|\mathbf{z}\|,\]  
where \(k_{M}=\max\{\|\mathbf{K}_{P}\|,\|\mathbf{K}_{D}\|,\|\mathbf{K}_{I}\|\}\). Thus, by recurring to the theory of perturbed systems [39], the effects of the two perturbation terms can be summarized as follows:  
* by virtue of Lemma 9.1 in [39], the origin of the system (39), under the vanishing perturbation is an exponentially stable equilibrium;
* by virtue of Lemma 9.2 in [39], the origin of the system (39), under the non-vanishing bounded perturbation is uniformly bounded. By exploiting the linearity of the nominal system (39), it can be stated that, under Assumption 1, the error \(\mathbf{z}\) is uniformly bounded. Moreover, if the planned acceleration vanishes after a time instant \(t_{f}\in\mathcal{T}\), the error converges to zero exponentially.