*3.2. Path Utility*

Path utility measures the relative effort needed for a robot to reach a task from its current location. The *path* utility function <sup>Ψ</sup>*<sup>i</sup>* : *<sup>T</sup>* <sup>×</sup> *<sup>S</sup>m*×*<sup>n</sup>* <sup>→</sup> [0, 1] is defined as follows:

$$\Psi\_i(T\_j, E\_{known}) = 2 \binom{\overline{\Delta} - \Delta\_i(T\_j)}{\overline{\Delta}}^\gamma \, \_{-1}^\gamma \tag{8}$$

s.t.

$$1 \le i \le M = |\mathcal{R}| , 1 \le j \le N = |T|.$$

where:

$$\begin{aligned} \overline{\Delta} &= \overline{d} - \underline{d} \\ \overline{d} &= \max \left\| X\_{i\cdot\cdot} T\_j \right\|\_{sp\ \prime} \forall j \\ \underline{d} &= \min \left\| X\_{i\cdot\cdot} T\_j \right\|\_{sp\ \prime} \forall j \\ \left. \Lambda\_i(T\_j) = \left\| X\_{i\cdot\cdot} T\_j \right\|\_{sp} - \underline{d} \right. \\ & \left. \left\| X\_{i\cdot\cdot} T\_j \right\|\_{sp} = \min\_{wp\_k \in E\_{kmnp}} \left\| x\_{k\cdot m} \right\|\_{\underline{k}} \sum\_{k=1}^{\ell-1} \left\| wp\_k - wp\_{k+1} \right\|\_{2} \right. \end{aligned}$$

Given the current environment knowledge *Eknown*, the function Ψ*<sup>i</sup>* estimates the path utility obtained by a robot *Ri* in case of selecting the task *Tj*. The parameter *γ* works as a shaping factor that could be used to tune the relation between distance and utility. The ordered sequence of waypoints *wpk* represents the shortest path between the robot configuration *Xi* and the target *Tj*. All segments (*wpk*, *wpk*<sup>+</sup>1) are safe given that they are always built regarding only the collision-free pathways present in the known region *Eknown*. The wavefront propagation method proposed by [43] is employed to determine the waypoint sequence. The shape and behaviour of the Ψ function are depicted in Figure 5.

**Figure 5.** Path utility function behaviour. There are several tasks in the scene (blue circles). The closest is located 6 m away from the robot while the furthest is 36 m far away. The closest and furthest tasks always return 1.0 and 0.0, respectively.
