*Preliminaries:*

1) Estimate a lower bound *M* , for instance *M* = *Mb* (Jordán & Bustamante, 2011),

2) Select a sampling time *h* as small as possible,

3) Choose design gain matrices *Kp* and *Kv* according to (56)-(57), and simultaneously in order to reduce *f* <sup>∗</sup> <sup>∆</sup>*Qn* and <sup>∆</sup>*Q*<sup>∗</sup> *tn* (see related commentary in previous section),

4) Define the adaptive gain matrices Γ*<sup>i</sup>* (usually Γ*<sup>i</sup>* = *αiI* with *α<sup>i</sup>* > 0),

5) Stipulate the desired sampled-data path references for the geometric and kinematic trajectories in 6 DOF´s: *ηrtn* and *vrtn* , respectively (see related commentary in previous section),

#### *Continuously at each sample point:*

6) Calculate the control thrust *τ<sup>n</sup>* with components *τ*1*<sup>n</sup>* in (29) and *τ*2*<sup>n</sup>* (40) (or (60)), respectively,

7) Calculate the adaptive controller matrices (44) with the lower bound *M* instead of *M*,

### *Long-term tuning:*

7) Redefine *Kp*, *Kv* and *h* in order to achieve optimal tracking performance.

#### **Remark**

For the present approach, we can summarize the different steps carried out in this Chapter after the control design in order to determine its convergence properties and performance of the control system:

a) Establishment of the adaptive laws for the designed controller using a lower bound of *M* (Section 6),

b) Stability and convergence analysis of the control system to a residual set dependent of the sign-undefinite terms *f*∆*Q*<sup>1</sup> and *f*∆*Q*<sup>2</sup> in (31) and (42), respectively, which depend on the pure dead-time *Td* (Section 7). Moreover, the conjoint incidence of local model errors *εηn*+<sup>1</sup> and *<sup>ε</sup>vn*<sup>+</sup><sup>1</sup> , measure noises *<sup>δ</sup>ηtn* and *<sup>δ</sup>vtn* , and the product *<sup>M</sup>*<sup>−</sup><sup>1</sup> *M* in the the convergence of state trajectories to a residual set B<sup>∗</sup> <sup>0</sup> is illustrated in (Section 7.1).

c) Proof of boundness of the adaptive controller matrices *Ui*'s and the way to ensure this (Section 7.2),

d) Analysis of a stability condition involving both huge sampling times and a large pure delays (Section 7.3).
