This work is aimed at the study of ways to assess risk in complex intelligent systems, which do not involve a human and focused mainly on issues related to the joint activity of unmanned systems in a dynamic environment, where risk appears due to their interaction. A simple and fast qualitative risk assessment method that can provide assessments reliably, efficiently and timely is proposed. The proposed method combines a matrix-based approach and a geometric approach to process numerical data at the input. A soft set is proposed as a suitable tool to avoid uncertainties related to incomplete and inaccurate information. The model of a dynamic system under risk based on n-dimensional stateful interaction space and corresponding topological structures is defined. It is proposed to estimate numerically only inexpensive parameters and then provide sophisticated quantification of them to avoid extensive calculations. The risk levels are assessed with respect to the certain cells of the discretized ndimensional interaction space. The levels of risk assessed in different cells of interaction space allow assessing risk distributed over the interaction space to prioritize risks properly. Due to uncertainty and lack of sufficient data, a density-based metric, which represents a relative density of interacting objects, is proposed to use instead of traditional probabilities or frequencies to estimate the chance of the object being exposed to the undesired event. The algorithm of the proposed risk assessment method is presented. The proposed method provides the acceptable performance of risk assessment enough to the real-time.
The world around us is a source of any risk. Most of the people’s usual activities are
risky. Often, people are forced to cope with risk in order to make a proper and
comprehensive decision in various ordinary activities such as driving a car, buying
expensive goods, lending from a bank, etc. In any case, to be responsible for the decisions,
people must be aware of their consequences, that is why many people use risk
asCopyright © 2020 for this paper by its authors. This volume and its papers are published under
the Creative Commons License Attribution 4.0 International (CC BY 4.0).
sessment techniques explicitly or implicitly [
Even though a person remains an important element of many complex systems since she manages them, makes important decisions, and, therefore, is responsible for their results, the modern level of technology development brings us new challenges. Unmanned systems are slowly but firmly and ruthlessly displacing people and taking full responsibility. Today, we are surrounded by lots of complex intelligent systems forced to make decisions on their own.
In human-based (ergatic) systems, the decision-maker mitigates the effect of subjectivity and uncertainty taking responsibility for the final result, but nobody can take control and responsibility in an autonomous unmanned system, so such a system must assess risk by itself. Thanks to intensive improvements, unmanned systems have become more and more intelligent and cheaper. This ensures the start of the massive use of unmanned systems that work together. Performing certain functions simultaneously, they interact with each other in a complex way entailing risk, which must be assessed reliably and timely.
It is not surprising, but even the most perfect system based on artificial intelligence
cannot be compared with a human in terms of the considered issues. Therefore,
serious challenges related to the limited resources of such systems, their limitations in the
computational performance, great sensitivity to incompleteness and inaccuracy of
information [
Thus, the study of ways to assess risk in complex intelligent systems, which do not involve a human, is a topic of our interest. In this paper, we focused mainly on issues related to the joint activity of unmanned systems in a dynamic environment, where risk appears due to their interaction. To assess risk in such conditions, we need a simple and fast method that can provide assessments reliably, efficiently, and timely.
Understanding of risk is domain-dependent, so its definitions differ essentially [
We consider dynamic systems that include a multitude of dynamic objects interacting in an uncertain and unpredictable environment. Since risk arises directly from the interaction of dynamic objects, the following assumptions can be introduced: • there is no risk if there is no dynamic and, consequently, there is no interaction; • there is a risk as soon as objects interact; • since the system is dynamic, the risk is also dynamic (i.e., it is not constant); • the risk could be applied to all participants in the interaction but varying degrees. Thus, emerging risk should not be ignored and must be assessed on time to provide a corresponding influence on decision-making at each interacting object. Since the risk is almost always present, it is necessary to assess its acceptability in a certain context.
Risk assessment is an important tool to minimize risks, and there are qualitative
and quantitative approaches to assess risk [
A quantitative approach has been proposed to estimate risk as a numerical value
based on the exposure, frequency (or probability), and consequence (i.e., the severity
of potential loss) [
A qualitative approach is most relied on the subjective judgment of the competent
person to determine an overall risk and can be mainly applied whenever the
quantitative risk assessment cannot be used for ethical reasons because the consequences are
harmful to people or death [
Let us make a small comparison. The qualitative approach is subjective but relatively simple. It less depends on exactly measured data than quantitative one because experts’ judgments supplement and clarify unavailable data.
The quantitative approach is very complex but more credible and more objective because they rely on meaningful statistics as well as on mathematical methods.
It is known that unmanned systems usually receive information about themselves
and the world around them using a variety of sensors. Some sensors provide
measurements, while others provide images, which require processing and analyzing. In
any case, all types of sensors can not provide complete information about the world
because they have limited accuracy [
There are several well-known and widespread risk assessment methods, which can
be used for this. Statistical methods [
Another well-known method is the matrix method [
There is a wide range of geometric methods based on the multidimensional representation of the state space of a certain system and the ranging of hazards using welldefined space and time limits (minima) [14, 15], which are relatively simple and based on quantitative calculations. However, most of them rely on a weak assumption of static limits and constant state changes.
Decision trees are used widely to assess risk [16] and have such advantages as well-applicability to combine the likelihood and impact of risks as well as flexibility in determining mitigation strategies. Their main disadvantages are a need for sufficient historical data and an inability for risk criticality assessment.
The Monte Carlo Simulation is another widespread risk assessment method [17] applicable to contingency modeling and generation of probability distributions but it requires a huge amount of data and it cannot establish an accurate estimate of the risks.
Artificial neural networks are widely used to quantify risks [18] and have some advantages since their inputs and outputs can be defined subjectively and do not need statistical distributions. Their disadvantages are mainly related to an inability to apply risk mitigation strategies.
Sensitivity analysis [19] is another method to assess risks that enables the definition of risk dimensions and ranking of hazards in their order of significance. However, it relies on a weak assumption that hazards arise one-at-a-time.
Fault tree analysis [20] provides experts a possibility to use linguistic terms rather than numerals to assess the probability of occurrence of hazards. Despite some advantages, this method depends only on experts’ opinions to analyze the hazards.
Fuzzy logic [21] is a method capable to deal with the vagueness and imprecision relevant to the qualitative risk assessment process but it also relies on the subjective opinions of experts in the definition of fuzzy membership functions.
Considering the advantages and disadvantages of the above-mentioned methods with respect to the objective of the research at hand, all methods that rely on statistics or the subjective opinion of experts are unacceptable since unmanned systems have neither statistics nor experts. Since the risk assessment in the context of our consideration is not concerned with obtaining precise outcomes, a qualitative assessment is quite acceptable at the output, but the main requirement is to prioritize risks in the order of their mitigation. At the same time, most of the available information is numerical but has limited accuracy and maybe blurred.
Thus, the use of the matrix method could be a good idea. It is also reasonable to
combine the matrix method with the geometric one for processing numerical
information at the input. Furthermore, a proper tool must supplement such a combination
to process incomplete and inaccurate information. Since complicated problems cannot
be solved using classical methods, there are several well-known tools to describe
various kinds of uncertainty, such as fuzzy sets [
Consider an abstract dynamic system Ω within a certain n -dimensional stateful space Ξ . Suppose the system Ω includes a set of m dynamic objects {ω1,...ωm} , each of which has an explicitly described state X m ∈ Ξ that can change in time. Suppose such objects ωi ∈ Ω can interact in a certain way within a given state space Ξ .
Suppose the n -dimensional space Ξ is Euclidean, linear, and uniform.
Let Y be a set of certain elements { y1,...yz} . Let T be an infinite set of time points t0 ,...ty ) strictly ordered by <T , t0 be an initial count, and ∆t is a time slice. Thus, (T ,t0 , ∆t, <T ) is a timescale defined over T .
Suppose ξT is a metric defined on T such that ti − t j T →τ endowed with the following properties: 1. ξT (ti ,t j ) = 0 ⇔ ti = t j ; 2. ξT (ti ,t j ) = ξT (t j ,ti ) ; 3. ξT (ti ,tk )„ ξT (ti ,t j ) + ξT (t j ,tk ) , ∀ti ,t j ,tk ∈ T .
Suppose a norm y = min ( y (t )) within space Ξ , where y ∈Y , t ∈T . Let us deΞ t∈[0,T ) fine a corresponding metric ξΞ ( y1, y2 ) = y1 − y2 endowed with the properties: 1. ξΞ ( y1, y2 ) = y1 − y2 =0 ⇔ y1 = y2 ; 2. ξΞ ( y1, y2 ) = y1 − y2 = y2 − y1 = ξΞ ( y2, y1 ) ; 3. ξΞ ( y1, y2 ) = ξΞ ( y1 + a, y2 + a) ; 4. ξΞ (λ y1,λ y2 ) = λ ⋅ξΞ ( y1, y2 ) where y1, y2, a ∈Y Suppose a certain basis e1,...,en holds uniformity of the metric ξΞ within n dimensional space Ξ . Thus, a certain state X i within space Ξ of the dynamic object ωi ∈ Ω can be described as Xi = ( x1i ,..., xni ) whereas x1i ,..., xni are the state parameters that correspond to the given basis e1,...,en .
Let us discretize space Ξ by a metric grid D using certain lines spaced with size δ . Suppose a linear mapping f : Ξ → D transforms the given space Ξ into a grid of n -dimensional cells of size δ in each n dimension.
As the result, we obtain the grid D = {dx,...z} of isometric n -dimensional cells dx,...z , where x,...z correspond to the cell state parameters x1,..., xn . In this way, each n dimensional cell dx,...z ∈ D is the smallest (discrete) subspace of the space Ξ . Therefore, the discrete state of each dynamic object ωi ∈ Ω is referenced to the corresponding cell within space Ξ . The size δ cell can usually be determined by the technical capabilities of sensors and the computing capabilities of onboard computers.
Suppose the proposed discretized model of space is consistent with the information captured by sensors. Let us build the corresponding topology.
Let D be a non-empty n -dimensional set of cells, R≥0 be a set of non-negative real numbers, and ξ D be a function D × D → R≥0 . Obviously, ξ D can be a suitable distance function (metric), if its values satisfy the conditions: 1. ξ D (d1, d1 ) = 0 if and only if d1 = d2 ; 2. ξ D (d1, d2 ) = ξ D (d2, d1 ) ; 3. ξ D (d1, d2 ) + ξ D (d2, d3 ) ≥ ξ D (d1, d3 ) for each d1, d2, d3 ∈ D . In this case, the function ξ D (d1, d2 ) =d1 − d2 provides a certain distance from a cell d1 to a cell d2 within the grid D , and a couple ( D,ξ D ) is a metric space.
Let ℜD ⊆ D × D be a reflexive, symmetric, and transitive relation defined on the n -dimensional set of cells D . It represents an indiscernibility relation ℜD (d1, d2 ) between two cells d1 and d2 such that d1, d2 ∈ D in terms of certain value y ∈Y , if (∀d1, d2 ∈ D)(∀y ∈Y ) y (d1 ) =y (d2 ) . In this case, the cells d1 and d2 are y indiscernible. Since the indiscernibility relation ℜD (d1, d2 ) is definitely an equivalence relation, it helps us to determine equivalence classes of D with respect to ℜD .
Suppose D / ℜD is a factor set that consists of all equivalence classes of D with respect to ℜD . Let D be a universal set and aprD =( D,ℜD ) be an approximation space defined by a composite set that is a finite union of elementary sets, each of which corresponds to an empty set or an element of the factor set. Thus, the equivalence class ℜD (d ) containing a specified cell d ∈ D clearly determines a family of all composite sets Def (aprD ) that, in turn, uniquely determines a topological space T = ( D, Def (aprD )) based on the approximation space aprD =( D,ℜD ) .
In this case, Def (aprD ) is a topology on D and T = ( D, Def (aprD )) is a corresponding topological space, if all subsets of the set Def (aprD ) satisfy the conditions: 1. ∅ ∈ Def (aprD ), D ∈ Def (aprD ) ; 2. A, B ∈ Def (aprD ) ⇒ A ∩ B ∈ Def (aprD ) ; 3. A, B ∈ Def (aprD ) ⇒ A ∪ B ∈ Def (aprD ) .
Certainly, each cell d ∈ D is an element of the topological space T .
Suppose each dynamic object ωi ∈ Ω performs a certain function changing its state Xi (t ) ∈ Ξ over time. The state of the system Ω is defined by a set of states of all dynamic objects {ω1,...ωm} ∈ Ω that constitute this system, so that X Ω (t ) = {Xi (t )}in=1 .
Since the state Xi (t ) of each dynamic object ωi ∈ Ω is uniquely determined by a correspondent cell dix,...iz of the grid D , as well as the overall state X Ω (t ) of the system Ω is determined by a set of cells {dix,...iz}in=1 , the topological space T based on D can be a relevant tool for risk assessment within space Ξ . 4
In the human-based system, a responsible person has to decide under risk rationally taking into account moral, ethics, and other reasons. In contrast, unmanned systems are based solely on feasibility expressed through the prism of given criteria. If there are many such criteria set simultaneously, then decision making will be multi-criteria.
Thus, the risk assessment is considered to be a very important task of a complex intelligent unmanned system. As we found out above, “no risk” situations are not mostly observed in the considered class of dynamical systems. Indeed, such a system is dynamic because its objects interact generating various threats and risks. Hence, for such a system, achieving the “near zero risk” is either impossible or too expensive. Thus, a much more important and achievable task for each of the dynamic objects is establishing an acceptable level of risk and use risk assessment methods to minimize risk during decision-making.
It should be noted that the dynamic object does not need to calculate the degree of risk quantitatively, but it needs to evaluate risk qualitatively in comparison to another risk to prioritize them. Therefore, even if there are ways to calculate risk parameters accurately, the use of complex time-consuming algorithms is not necessary.
In this paper, it is proposed to use a simple and fast qualitative method based on a risk matrix. The risk matrix should correspond to the available data and the exposure, frequency (probability, possibility), and consequences (severity of potential loss). Quantitative risk assessment requires numerical estimations of such parameters.
We propose to estimate numerically only inexpensive parameters and then provide sophisticated quantification of them to avoid extensive calculations. Such quantified values can then be used as inputs to assess a risk level at the output based on the risk matrix. In this way, we assess the risk level with respect to the certain cells dix,...iz of the discretized interaction space D . Ultimately, knowing the levels of risk in different cells of space, we can assess risk distributed over this space to prioritize risks properly. To mitigate the effect of quantification, we use soft sets, which allow us to construct a distribution of risk levels over the space D . 4.1
The Algorithm of the Risk Assessment
The risk assessment is a continuous process that includes (Fig. 1): • determination of the risk context; • identifications of threats; • risk estimation: 1. estimation of exposure; 2. estimation of possibility; 3. estimation of consequences; • risk evaluation; • risk prioritization.
Determination of the risk context is aimed at analyzing possible interactions between dynamic objects and evaluating all three components necessary for the risk assessment.
At the next stage, some of these interactions, which can pose risk, are identified as threats. Since threats can cause injury, damage, or loss, the purpose is to identify as many threats as possible. Threats Identification
Risk Estimation Severity Density
Exposure Risk Evaluation
Risk is Tolerable?
No Risk Prioritization
Risk
Analysis Yes
Risk
Assessment At the next stage, the qualitative assessment is provided to determine the level of risk associated with each specific threat identified at the previous stage. Usually, this stage is based on the estimations of probability, exposure, and severity of loss as the consequences of an undesirable event. In our case, the exposure and severity can be estimated using the interaction analysis, but the probability can not be estimated due to a lack of sufficient data, so we propose to use density estimations instead of probability estimations.
Finally, risks must be evaluated and prioritized to reduce, mitigate, or eliminate them in the context of all components. Since such components of risk (e.g., density, exposure, and severity) usually cannot be identified unequivocally, we use quantified levels that can be estimated using ordered scales of limits. Thus, in our case, the risk assessment can be reduced to a matrix based on levels of all three components. Suppose dynamic objects ωi ,ω j ∈ Ω interact within space D during their joint activity. The trajectory Tr (ωi ) of each object ωi can be represented as a continuous sequence of its states Xi (t j ),...Xi (tk ) on a time interval t j ,...tk ∈ T , while the state X i (t ) of the ωi at the moment t ∈ T corresponds to the certain cell dix,...iz ∈ D determined by its parameters within D .
The objects, which states are within the interaction space D do not necessarily interact. Their trajectories at different time points can approach, intersect, or diverge from each other. Basically, their interaction can be defined based on the local proximity D∗ of the dynamic objects within the interaction space D , which can be estimated based on the given metric ξ D (Fig. 2).
Thus, D∗ is a subspace of the interaction space D , D∗ ⊆ D , usually represented by a closed n -dimensional figure having the maximum surface distance R from a certain base point d0 such that d ∈ D∗ ⇔ ξ D ( d0 , d )„ R .
e3 О ω1 e2 ω2 ω0(t0)
Dt∗0
Tr (ω0 ) ω0(t1)
e1 Let c (Ω, Ω) be a relation between two dynamic objects ωi ,ω j ∈ Ω that restricts a subset of interacting objects Ωc , i.e. {ωi,ω j} ∈ Ωc ↔ c (ωi,ω j ) . Due to the dynamics of the interaction, the composition Ωc also changes dynamically.
Let ri (t ) = {ri0 (t ),...ril (t )} be a set of n -dimensional margins evaluated for a certain object ωi at the time t based on the above-mentioned metric ξ D such that rij (t ) = rij1 (t ),...rijn (t ) . Since the trajectory Tr (ωi ) of each object ωi is characterized by the different dynamic (e.g., speed and acceleration) and functional parameters (e.g., prudence or persistence), each of the interacting objects will simultaneously have its own set of margins that can differ from the set of margins for other objects.
Suppose a function Χ (ωi ,t ) returns the state X i (t ) of the object ωi at the time t as well as the function Χ (ωk ,t ) returns the state X k (t ) of the object ωk at this time. Thus, a certain distance ∆ (ωi ,ωk ) between their trajectories can be evaluated by Clearly, the distance ∆ (ωk ,ωi ) with respect to the object ωk ∆ (ωi ,ωk ) .
Assume that unpleasant consequences are concerned with an excessive rapprochement or intersection of the trajectories of interacting objects ωi and ωk .
Thus, we define interaction function c i : D → rij , which describes a subset of objects Ωc i ⊆ Ω interacting with ωi , as a surjective anisometric mapping based on the evaluated sets of n -dimensional margins ri (t ) and rk (t ) (Fig. 3).
Suppose τ ik (t ) = {τ 0 (t ),...τ q (t )} is a set of time limits given with respect to the interaction of ωi and ωk based on the metric ξT over T such that ti − tk T →τ .
ω2 D* ξ D r02 ω0 c i (ωi ,ω j ) ), ∀ωi ,ωk c i > ri1 ⇔ ωi ,ωk ∈ A c i .
The dynamic
object Χ (ωi ,t ) − Χ (ωk ,t ) ≥ ri2 (t ) , Χ (ωi ,t ) − Χ (ωk ,t ) ≥ ri3 (t ) . Let ґ r be a partial order ri0 (t ) ґ r ri1 (t ) ґ r ... ґ r riz (t ) given on the set ri (t ) that implies the corresponding ordinal scale Ri (t ) = {ri0 (t ),...riz (t )} . Suppose µi is a function that uniquely takes each distance ∆ (ωi ,ωk ) onto a certain level of the scale Ri (t ) , such that µi : Χ (ωi ,t ) − Χ (ωk ,t ) → rij (t ) ∈ Ri (t ) . Thus, the function µi can be used for quantification of the states’ distances onto the scale Ri (t ) .
The dynamic object ωi interacts with ω j (in other words, there is a relation if and only if Χ (ωi ,t ) − Χ (ωk ,t ) ≥ ri1 (t ) .
Consequently, ωi and interacts interacts with with ω j ω j dangerously, critically, if if
Thus, the interaction set Ωc i around ωi includes all objects ω j ∈ Ω, ∀j ≠ i , which have established the interaction with ωi based on c i .
Clearly, the margin limits outline n -dimensional areas within subspace D∗ , which represent exposure levels needed for the risk assessment. Definitely, all objects that interact with the object ωi (i.e., their distances are equal or greater than ri1 (t ) ) pose risk to ωi , so the object ωi is exposed to risk.
Let us use margin limits as it is shown in Table 1. At the same time, severity levels can be evaluated based on the inertial properties of the objects within the interaction space D . Indeed, to avoid the intersection of trajectories objects can speed up, slow down, or deviate, while the environment can both increase the inertial force or weaken it. Thus, the time-to-intersect τ ik (t ) can be a relevant measure, which makes it possible to assess how great is the influence of the dynamics of the objects’ ωi and ωk interaction on its possible consequences.
Let ґ t be a partial order τ ik0 (t ) ґ t τ ik1 (t ) ґ t ... ґ t τ ikq (t ) given on the set τ ik (t ) that implies the corresponding ordinal scale U ik (t ) = {τ ik0 (t ),...τ ikq (t )} . Suppose ηik is a function that uniquely takes time-to-intersect τ ik (t ) onto a certain level of the scale U ik (t ) , such that ηik :τ ik (t ) →τ ikj (t ) ∈U ik (t ) . Thus, the function ηik can be used for quantification of the remaining time during the interaction onto the scale U ik (t ) .
Based on the time-to-intersect τ ik (t ) and the scale U ik (t ) , the interaction space D can be divided into the subspaces of easy activity Di0k , restricted activity Di1k , desired deviation Di2k , and obligatory deviation Di3k , which can be outlined around the state X (ωi ,t ) of the object ωi ∈ Ωc at the time t , i.e., around the cell that reflect the current state of the object ωi within D .
Naturally, the less time it takes to prevent an undesirable event, the less is the possibility of preventing such event and the more serious are its consequences. Hence, we can define the severity levels based on the time-to-intersect limits.
Let us use time limits as it is shown in Table 2. The next and last component of risk assessment is a chance to the object of being exposed to the unpleasant event and its consequences. Usually, this chance can be accurately estimated using the probability of an event or the frequency of occurrence of an unpleasant event. However, as shown above, it is not always possible to obtain such estimates. Moreover, in the considered class of dynamic systems, the statistics of the occurrence of undesirable events are unavailable due to the non-representativeness of the sample. Therefore, we propose to use density-based metrics to estimate the chance of the object being exposed to an unpleasant event. The relative density of interacting objects is the simplest density metric that can be easily applied to the risk assessment. Consider the interaction set Ωc i for the object ωi . This set includes all objects that interact with ωi based on the relation c i . These objects are dispersed over the subspace D∗ of the interaction space defined as the closed n -dimensional area described by the maximum surface distance R starting from the point d0 , which can be spatially aligned with the cell di that represents the state X i (t ) of the object ωi within the interaction space D .
Obviously, the higher is the density of objects within the subspace D∗ , the higher is the likelihood of an unpleasant event. Therefore, the relative density of interacting objects si (t ) with respect to the object ωi can be estimated as the ratio of the number of interacting objects N that constitute the interaction set Ωc i to the relative volume of the subspace D∗ , which can be obtained as the number of cells within the grid D that are inscribed in the subspace D∗ . Consequently, si (t ) = Ωc i D∗ , where ⋅ is the cardinality of the corresponding set.
Let ґ s be a partial order σ i0 (t ) ґ s σ i1 (t ) ґ s ... ґ s σ iu (t ) given on the set si (t ) that implies the corresponding ordinal scale Si (t ) = {σ i0 (t ),...σ iq (t )} . Suppose γ i is a function that uniquely takes the relative density si (t ) onto a certain level of the scale Si (t ) , such that γ i : si (t ) → σ ij (t ) ∈Si (t ) . Thus, the function γ i can be used for quantification of the relative density of interacting objects within the interaction space D .
We can use a limited number of density levels to speed up the risk assessment as is shown in Table 3.
0 1 2 3 Thus, we have three components of risk represented qualitatively by certain levels and we are ready to assess the risk level. The goal of risk assessment is to determine risk acceptability, often by comparison to similar risks. Thus, risk assessment is relative. Estimated levels of all three components of risk intersected by rows and columns create a Risk Assessment Matrix. Risk Assessment Matrix is a generally accepted definition for risk levels given in Table 5. Soft sets enable to approximately represent risk levels distributed over the interaction space using the topologies defined in Section 3.
Let yi be an i -th risk level and Y = { yi}i5=0 be an ordered set of risk levels. Suppose D is a universe and ϒ is a mapping that takes Y onto a set of all subsets of the set D , i.e. ϒ : yi → 2D . Thus, a pair ( ϒ,Y ) represents a soft set of cells [24]. In other
X - Extreme
Z I I I Z I I I I I L L I L L L
Z M I M I M L H I M L M L H L H L M L H M H M H L H M H M X M X
H H H H H H H X H X X X X X X X words, the pair (ϒ,Y ) is a family of subsets of the set of cells D , all of which are parameterized by the set Y . Each value yi ∈Y defines a certain set of yi approximated elements of the soft set (called yi -elements of the soft set [25]). Let us denote such yi -elements by ϒi .
Soft set (ϒ,Y ) divides the universe D into the set of yi -elements such that ϒ = ∪{ϒi }ik=1 . A dynamic yi -indiscernibility relation ℜDyi (t ) can be defined on the universe D as (∀yi ∈Y ) ℜDyi (t ) ={(dm , dn ) ∈ D × D yi (dm ,t ) =yi (dn ,t )} . Using this relation, each yi -element of the soft set ϒi represents a certain equivalence class at the moment t . Thus, the parameterized family of subsets of the universe D constitutes the yi -element of the set ϒi , which uniquely determines a factor-set D / ℜDyi (t ) . The factor-set consists of all equivalence classes of D induced by the relation ℜDyi (t ) . Therefore, a pair aprD =( D,ℜDyi (t )) defines the dynamic approximation space. Definitely, Def (aprD ) is a family of all compound sets and T DℜDi (t ) = ( D, Def (aprD )) is a y dynamic soft topological space, which uniquely corresponds to the dynamic approximation space.
Each yi -element of the soft set ϒi enumerates cells, which corresponds to the certain i -th risk level. Thus, different yi -elements of the soft set have distinctive risk levels. The cells, which belong to the yi -element of the soft set ϒi , constitute the topologic structure represented by the corresponding equivalence class D / ℜDyi (t ) . Since the assessments of risk usually change over time, the soft set (ϒ,Y ) of cells is dynamic as well as risk assessments. The representation of the dynamic soft set of risk assessments distributed over the interaction space is shown in Fig. 4. The proposed method has been tested in the unmanned vehicle’s onboard system Breeze [26] based on embedded microcontroller STM32F429 (180 MHz Cortex M4, 2Mb Flash/256Kb RAM internal, QSPI Flash N25Q512). The proposed algorithm of the risk assessment has been implemented using the C++ programming language as well as the ToPo and SoFTo library, which offer a set of operations for topologies including their addition and subtraction, determining unions, intersections, closures, and interiors, and a wide range of operations with soft sets.
The efficiency of the proposed method has been examined based on its comparison with traditional geometric (quantitative) and decision-tree based (qualitative) risk assessment methods. The total time of the risk assessment has been evaluated with respect to the number of interacting dynamic objects varied from 10 to 100. The results are shown in Fig. 5, they show that the proposed method provides acceptable performance, which allows it to be used in real-time unmanned systems. The problem of risk assessment in complex intelligent systems, which do not involve a human is addressed in the paper. The paper is focused on issues related to the joint activity of unmanned systems in a dynamic environment, where risk appears due to their interaction. The authors propose the method that combines a matrix-based approach and a geometric approach to process numerical data at the input and uses soft sets as a suitable tool to avoid uncertainties related to incomplete and inaccurate information. The proposed method is based on the model of a dynamic system under risk based on n-dimensional stateful interaction space and corresponding topological structures. Authors propose to estimate numerically only inexpensive parameters and then provide their quantification to avoid extensive calculations. The risk levels are assessed with respect to the certain cells of the discretized n-dimensional interaction space. The levels of risk assessed in different cells of interaction space allow assessing risk distributed over the interaction space to prioritize risks properly. Due to uncertainty and lack of data, a density-based metric, which represents a relative density of interacting objects, is proposed to use instead of traditional probabilities or frequencies to estimate the chance of the object being exposed to the undesired event.
The proposed qualitative method is simple and fast, it provides the acceptable performance of risk assessment enough to the real-time unmanned systems.