## 4 Adaptive \(\alpha\)-Value Computation
### [Cl/Co,F/Co] and [F/Co,Cl/NC] Cases
#### 4.3.1 [F/Co,Cl/NC] case.
Finally, the expression for an _HO-Threshold_ dependent \(\alpha\), say \(\alpha_{\text{HO}}\), is expressed in (15) as follows:  
\[\alpha_{\text{HO}}=\frac{\Omega_{1}}{1-\Psi_{i}(T_{\text{HO}})+\Omega_{1}} \tag{15}\]  
**Proposition 3**. : _The applicability of the \(\alpha_{\text{HO}}\) referred to in (15) causes any task within the threshold scope that also offers any positive connectivity level to be favoured over the rest of the tasks that do not offer any connectivity level, regardless of how close to the robot they are._  
Proof. : \(\Phi_{i}(\alpha,T_{j},R)\geq\Phi_{i}(\alpha,T_{k},R)\) is imposed to any tasks \((T_{j},T_{k})\) that respect the [F/Co,Cl/NC] conditions:  
\[\Phi_{i}(\alpha,T_{j},R)=\alpha\cdot\Psi_{i}(T_{j})+\beta\cdot \Omega_{i}(T_{j},R) \geq\alpha\cdot\Psi_{i}(T_{k})=\Phi_{i}(\alpha,T_{k},R)\] \[\alpha\cdot\Psi_{i}(T_{j})+(1-\alpha)\cdot\Omega_{i}(T_{j},R) \geq\alpha\cdot\Psi_{i}(T_{k})\] \[\alpha\cdot(\Psi_{i}(T_{j})-\Omega_{i}(T_{j},R))+\Omega_{i}(T_{j },R) \geq\alpha\cdot\Psi_{i}(T_{k})\] \[\frac{\Omega_{i}(T_{j},R)}{\overline{\Psi_{i}(T_{k})-\Psi_{i}(T_ {j})+\Omega_{i}(T_{j},R)}} \geq\alpha\]  
Then, applying (15) leads to (16):  
\[\frac{\Omega_{i}(T_{j},R)}{\overline{\Psi_{i}(T_{k})}-\overline{ \Psi_{i}(T_{j})}+\Omega_{i}(T_{j},R)}\geq\frac{\Omega_{1}}{1+\Omega_{1}-\Psi_ {i}(T_{\text{HO}})}\] \[\Psi_{i}(T_{j})\geq\frac{\Omega_{i}(T_{j},R)}{\Omega_{1}}\cdot( \Psi_{i}(T_{\text{HO}})-1)+\Psi_{i}(T_{k}) \tag{16}\] \[\Psi_{i}(T_{j})=sup\left(\frac{\Omega_{i}(T_{j},R)}{\Omega_{1}} \cdot(\Psi_{i}(T_{\text{HO}})-1)+\Psi_{i}(T_{k})\right)\]  
Since _i_) \(\Omega_{1}\) is constant, _ii_) \(\frac{\Omega_{i}(T_{j},R)}{\Omega_{1}}\geq 1\), and _iii_) \((\Psi_{i}(T_{\text{HO}})-1)\leq 0\), it is possible to conclude that:  
* \(\Psi_{i}(T_{j})\) is monotonically decreasing concerning \(\Omega_{i}(T_{j},R)\). * the upper bound is reached when: 1. \(\Psi_{i}(T_{\text{HO}})=1\) 2. \(0\leq\Psi_{i}(T_{\text{HO}})<1\), \(\Omega_{i}(T_{j},R)=\Omega_{1}\) and \(\Psi_{i}(T_{k})=1\). Please note that (a) is out of the proposition conditions. Instead, (16) can be rewritten imposing (b), leading to:  
\[\Psi_{i}(T_{j}) \geq\frac{\Omega_{1}}{\Omega_{1}}\cdot(\Psi_{i}(T_{\text{HO}})-1)+1)\] \[\alpha_{\text{HO}}=\frac{\Omega_{1}}{1-\Psi_{i}(T_{\text{HO}})+\Omega_{1}} \tag{15}\]0  
which is true if, and only if, \(\Delta_{i}(T_{j})\leq\text{HO-Threshold}\), which is indeed what the human operator would like to get from his criterion application to tasks within the _HO-Threshold_. Hence, following (15) under the [F/Co,Cl/NC] conditions it is always possible to compute an \(\alpha_{\text{HO}}\)-value that makes the robots behave following the human-operator criterion. Likewise, it is important to highlight that the \(\alpha_{\text{HO}}\)-value needs to be calculated every time a robot is ready to make a decision. This need for adaptation arises from \(\Psi_{i}(T_{\text{HO}})\), which is not constant. Its value depends on the relation between the _HO-Threshold_ and the relative distance to the current furthest task. That way, the robots can autonomously adapt the weights of the _task_ utility function according to the changing conditions of the environment in order to be always consistent with the human-operator criterion. \[\alpha_{\text{HO}}=\frac{\Omega_{1}}{1-\Psi_{i}(T_{\text{HO}})+\Omega_{1}} \tag{15}\]1 \[\alpha_{\text{HO}}=\frac{\Omega_{1}}{1-\Psi_{i}(T_{\text{HO}})+\Omega_{1}} \tag{15}\]2 (20) s.t. \[\alpha_{\text{HO}}=\frac{\Omega_{1}}{1-\Psi_{i}(T_{\text{HO}})+\Omega_{1}} \tag{15}\]3 \[\alpha_{\text{HO}}=\frac{\Omega_{1}}{1-\Psi_{i}(T_{\text{HO}})+\Omega_{1}} \tag{15}\]4 given that: \[\alpha_{\text{HO}}=\frac{\Omega_{1}}{1-\Psi_{i}(T_{\text{HO}})+\Omega_{1}} \tag{15}\]5 \[\alpha_{\text{HO}}=\frac{\Omega_{1}}{1-\Psi_{i}(T_{\text{HO}})+\Omega_{1}} \tag{15}\]6  
On this domain, the function \(\frac{x}{u+x}\) presents an absolute maximum equal to 1 in the point \((x,u)=(1,0)\), and absolute minima equal to 0 along the line segment defined by \((x,u)=(0,u)\).