**7. Conclusion**

i = 1.

part CF to 8.30 a.m. are taken into account.

the shorter one BCF (with minimal risk) is chosen.

back to the initial point B of the part is being.

the shorter one BGKF (with minimal risk 0.051) is chosen.

is equal to 0.067, for the route GKF risk equals 0.083).

for routes BCF, BGKF, and BGHLDEF.

part of the route is not overcome).

*Probability, Combinatorics and Control*

i = i + 1 = 2.

**Figure 18.**

*(Finish).*

**28**

**Step 2** (i = 1). Using probabilistic model, a calculation of the probability of failure is carried out for each variant. From the set of variants ABCF, AGKF, and AHLDEF, the shorter variant ABCF for which risk is equal to 0.034 is chosen (for the route AGKF risk = 0.051, for route AHLDEF risk = 0.067), see **Figure 18**. The relevant data from the drone about the forecasted conditions and the weather on the

**Step 3** (i = 1). The robot overcomes the part AB of route. For the new initial point B, the input for modeling every part of possible route is updated in real time

**Step 4** (i = 1). The robot has not yet arrived at the intended point F (i.e., the last

**Step 2** (i = 2 for variants BCF, BGKF, and BGHLDEF). Input for modeling is not changed. Risks are the same. From the route variants BCF, BGKF, and BGHLDEF,

**Step 3** (i = 2 for variant BCDEF). The robot overcomes the part BC. For the new initial point C, the input for modeling every part of possible route is updated in real time: bad weather on the CF part does not allow further movement. And weather improvements in the next 2 h are not expected. Part CF is impassable. The come-

**Step 2** (i = 2 for two remaining variants). From variants BGKF and BGHLDEF,

**Step 3** (i = 2 for variant BGKF). The robot overcomes the part BG. For the new initial point G, the input for modeling every part of possible route is updated in real time: according drone from 9.00 a.m. on parts GK and KF the imminent avalanche are detected. The accumulated knowledge is used to clarify the input for modeling, namely: the frequency threats in the part GKF increases from 1.5 to 2.5 times at 10 h. Using a probabilistic model for each variant, a recalculation of the risk of failure is carried out. Of the variants GKF and GHLDEF, the variant GHLDEF is chosen (risk

*The risk of "failure" in dependence on prognostic period during the robot route from point A (Start) to point F*

The proposed approach to build and implement the probabilistic methods and models is demonstrated by application to cognitive solving:


There is proposed to carry out probabilistic prediction of critical processes in time so that not only to act according to the prediction, but also to compare predictions against their coincidence to the subsequent realities.

The described analytical solutions are demonstrated by practical examples such as:

System planning the possibilities of functions performance in space by using robot-manipulators, by AIS for a coal company and for a floating oil and gas platform

Forming input for probabilistic modeling from monitored data Robot route optimization under limitations on risk of "failure" in conditions of uncertainties

A cognitive solving of the chosen problems consists in improvements, accumulation, analysis, and use of appearing knowledge.

## **Appendix**

Proofs for formulas (1)–(3)

According to the proof of formula (1): because between diagnostics system is not protected from threats an influence (a loss of integrity) will take place only after danger occurrence and activation during given time before the next diagnostic (**Figure 6**). A risk to lose integrity (i.e., probability of "failure") is equal to

Ωpenetr\*Ωactiv(Treq) because these PDF are independent. The found probability of providing system integrity (probability of "success") is equal to addition to 1.

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The proof of formula (1) is complete.

For the special case, if Ωoccur(t) = 1 � exp(σt), σ = 1/Toccur, Ωactiv(t) = 1 � exp (t/β), β = Tactiv

$$P\_{(1)}\left(T\_{\text{given}}\right) = \begin{cases} \left(\sigma - \beta^{-1}\right)^{-1} \left\{\sigma e^{-T\_{\text{given}}/\beta} - \beta^{-1} e^{-\sigma T\_{\text{given}}}\right\}, \text{if } \sigma \neq \beta^{-1}, \\\ e^{-\sigma T\_{\text{given}}} \left[1 + \sigma T\_{\text{given}}\right], \text{if } \sigma = \beta^{-1}. \end{cases}$$

Note. This formula (1) is used also for the estimation of system operation without diagnostics. There is supposed that before the beginning of period *Tgiven* system integrity is provided.

According to the proofs of formulas (2) and (3), we consider independence. Then formula (2) means measure *P*(2)(Tgiven) = *P*mdl + *P*end, where *P*mdl is the probability of correct operation ("success") within the period Tgiven since beginning to the last diagnostics, *P*mdl = N((Tbetw + Tdiag)/Treq)*P*(1) N(Tbetw + Tdiag), here *P*(1)(Tbetw + Tdiag) is defined by formula (1), but one is calculated only for time Tbetw + Tdiag; *P*end is the probability of correct operation ("success") within the assigned period Tgiven after the last diagnostics, i.e. in the last remainder Trmn = Treq – [N(Tbetw + Tdiag)], *P*end = (Trmn/Treq) *P*(1)(Trmn). Here, *P*(1)(Trmn) is defined by formula (1), but one is calculated only for the remainder time Trmn. Really, for this time Trmn, the main condition of the first variant is true: Trmn < Tbetw + Tdiag.

Formula (3) means measure *P*(2)(Tgiven) = *P*(1) N(Tbetw + Tdiag)*P*(1)(Trmn). Interpretation is the next: "success" is on all N periods (Tbetw + Tdiag) AND on remainder time Trmn.

The proofs for formulas (1)–(3) are complete.
