Let consider that both Target and Object, a subject of mutual navigation, to be presented as octagons, depicted on Figure 1 [1] . Also, we use octagons for simplification’s sake only. Given the fact that we are studying a projection-based model, both targets and objects could be presented as follows [1] :

T = { t j } ; j = 1 , n ¯ . Where j is number of heights of a Target, whereas O = { o i } ; i = 1 , m ¯ and i is number of heights of an Object. Both a target and an object could be presented in three-dimensional space as follows:

t j ∈ T = { x j t , y j t , z j t } ; j = 1 , n ¯ ,     o j ∈ O = { x i o , y i o , z i o } ; i = 1 , m ¯ . (2.1)

On the other hand, from Figure 1 each value of both a Target and an Object coordinate could be presented as a pair of minimal and maximal (per 3D coordinate) values of them. For targets, in particular

∀ j ∈ [ 1 , n ] | x min T = min j { x j t } , x max T = max j { x j t } , y min T = min j { y j t } , y max T = max j { y j t } , min j { z j t } , z max T = max j { z j t } , z min T = min j { z j t } (2.2)

By analogy, for objects we are getting:

∀ j ∈ [ 1 , n ] | x min O = min j { x j o } , x max O = max j { x j o } , y min O = min j { y j o } , y max O = max j { y j o } , min j { z j o } , z max O = max j { z j o } , z min O = min j { z j o } (2.3)

Considering (2.2)-(2.3) we can formulate some logical predicates, which would describe mutual positioning of two players in the paradigm of a projection-based model. Let us define predicates as relation symbols, describing a variety of positions of two entities in a space in a connection to each other.

Before formulation of a possible actions, which could be performed by certain entities, and given (2.2) and (2.3) we have to consider for each entity the following points in 3-dimentional space T c e n t e r = { x c e n t e r T , y c e n t e r T , z c e n t e r T } for a Target and O c e n t e r = { x c e n t e r O , y c e n t e r O , z c e n t e r O } for an Object correspondingly, These points could define some conditional center of a gravity for each of them (median points in space)

x c e n t e r T = x max T + x min T 2 , x c e n t e r O = x max O + x min O 2 (2.4)

y c e n t e r T = y max T + y min T 2 , y c e n t e r O = y max O + y min O 2 (2.5)

z c e n t e r T = z max T + z min T 2 , z c e n t e r O = z max O + z min O 2 (2.6)

We define the following predicates [2] by using (2.4)-(2.6)

1) Object docks in front of a Target (DIF)

D I F ( O , T ) ⇒ x c e n t e r T = x c e n t e r O & z c e n t e r T = z c e n t e r O & y max T = y min O (2.7)

Δ x . . = x c e n t e r T . . − x c e n t e r O . . (2.8)

c l X = { ( x c e n t e r T . . / x c e n t e r O . . ) ∗ 100 , Δ x ≤ 0 , ( x c e n t e r O . . / x c e n t e r T . . ) ∗ 100 , othervise (2.9)

Δ y . . = y min O . . − y max T . . (2.10)

c l Y = { ( y min O . . / y max T . . ) ∗ 100 , Δ y ≤ 0 , ( y max T . . / y min O . . ) ∗ 100 , othervise (2.11)

Δ z . . = z c e n t e r T . . − z c e n t e r O . . (2.12)

c l Z = { ( z c e n t e r T . . / z c e n t e r O . . ) ∗ 100 , Δ z ≤ 0 , ( z c e n t e r O . . / z c e n t e r T . . ) ∗ 100 , othervise (2.13)

2) Object docks at back of a Target (DAB)

D A B ( O , T ) ⇒ x c e n t e r T = x c e n t e r O & z c e n t e r T = z c e n t e r O & y min T = y max O (2.14)

Δ x . . = x c e n t e r T . . − x c e n t e r O . . (2.15)

c l X = { ( x c e n t e r T . . / x c e n t e r O . . ) ∗ 100 , Δ x ≤ 0 , ( x c e n t e r O . . / x c e n t e r T . . ) ∗ 100 , othervise (2.16)

Δ y . . = y max O . . − y min T . . (2.17)

c l Y = { ( y max O . . / y min T . . ) ∗ 100 , Δ y ≤ 0 , ( y min T . . / y max O . . ) ∗ 100 , othervise (2.18)

Δ z . . = z c e n t e r T . . − z c e n t e r O . . (2.19)

c l Z = { ( z c e n t e r T . . / z c e n t e r O . . ) ∗ 100 , Δ z ≤ 0 , ( z c e n t e r O . . / z c e n t e r T . . ) ∗ 100 , othervise (2.20)

3) Object docks at left of a Target (DAL)

D A L ( O , T ) ⇒ y c e n t e r T = y c e n t e r O & z c e n t e r T = z c e n t e r O & x min T = x max O (2.21)

Δ x . . = x min T . . − x max O . . (2.22)

c l X = { ( x max O . . / x min T . . ) ∗ 100 , Δ x ≤ 0 , ( x min T . . / x max O . . ) ∗ 100 , othervise (2.23)

Δ y . . = y c e n t e r T . . − y c e n t e r O . . (2.24)

c l Y = { ( y c e n t e r T . . / y c e n t e r O . . ) ∗ 100 , Δ y ≤ 0 , ( y c e n t e r O . . / y c e n t e r T . . ) ∗ 100 , othervise (2.25)

Δ z . . = z c e n t e r T . . − z c e n t e r O . . (2.26)

c l Z = { ( z c e n t e r T . . / z c e n t e r O . . ) ∗ 100 , Δ z ≤ 0 , ( z c e n t e r O . . / z c e n t e r T . . ) ∗ 100 , othervise (2.27)

4) Object docks at right of a Target (DAR)

D A R ( O , T ) ⇒ y c e n t e r T = y c e n t e r O & z c e n t e r T = z c e n t e r O & x max T = x min O (2.28)

Δ x . . = x max T . . − x min O . . (2.29)

c l X = { ( x min O . . / x max T . . ) ∗ 100 , Δ x ≤ 0 , ( x max T . . / x min O . . ) ∗ 100 , othervise (2.30)

Δ y . . = y c e n t e r T . . − y c e n t e r O . . (2.31)

c l Y = { ( y c e n t e r T . . / y c e n t e r O . . ) ∗ 100 , Δ y ≤ 0 , ( y c e n t e r O . . / y c e n t e r T . . ) ∗ 100 , othervise (2.32)

Δ z . . = z c e n t e r T . . − z c e n t e r O . . (2.33)

c l Z = { ( z c e n t e r T . . / z c e n t e r O . . ) ∗ 100 , Δ z ≤ 0 , ( z c e n t e r O . . / z c e n t e r T . . ) ∗ 100 , othervise (2.34)

5) Object docks on top of a Target (DOT)

D O T ( O , T ) ⇒ x c e n t e r T = x c e n t e r O & y c e n t e r T = y c e n t e r O & z max T = z min O (2.35)

Δ x . . = x c e n t e r T . . − x c e n t e r O . . (2.36)

c l X = { ( x c e n t e r T . . / x c e n t e r O . . ) ∗ 100 , Δ x ≤ 0 , ( x c e n t e r O . . / x c e n t e r T . . ) ∗ 100 , othervise (2.37)

Δ y . . = y c e n t e r T . . − y c e n t e r O . . (2.38)

c l Y = { ( y c e n t e r T . . / y c e n t e r O . . ) ∗ 100 , Δ y ≤ 0 , ( y c e n t e r O . . / y c e n t e r T . . ) ∗ 100 , othervise (2.39)

Δ z . . = z max T . . − z min O . . (2.40)

c l Z = { ( z min O . . / z max T . . ) ∗ 100 , Δ z ≤ 0 , ( z max T . . / z min O . . ) ∗ 100 , othervise (2.41)

6) Object docks under (at bottom) of a Target (DUN)

D U N ( O , T ) ⇒ x c e n t e r T = x c e n t e r O & y c e n t e r T = y c e n t e r O & z min T = z max O (2.42)

Δ x . . = x c e n t e r T . . − x c e n t e r O . . (2.43)

c l X = { ( x c e n t e r T . . / x c e n t e r O . . ) ∗ 100 , Δ x ≤ 0 , ( x c e n t e r O . . / x c e n t e r T . . ) ∗ 100 , othervise (2.44)

Δ y . . = y c e n t e r T . . − y c e n t e r O . . (2.45)

c l Y = { ( y c e n t e r T . . / y c e n t e r O . . ) ∗ 100 , Δ y ≤ 0 , ( y c e n t e r O . . / y c e n t e r T . . ) ∗ 100 , othervise (2.46)

Δ z . . = z min T . . − z max O . . (2.47)

c l Z = { ( z min T . . / z max O . . ) ∗ 100 , Δ z ≤ 0 , ( z max O . . / z min T . . ) ∗ 100 , othervise (2.48)

We represent clX from (2.9), (2.16), (2.23), (2.30), (2.37), (2.44), and also clY from (2.11), (2.18), (2.25), (2.32), (2.39), (2.46) and clZ from (2.13), (2.20), (2.27), (2.34), (2.41), (2.48) as a fuzzy set, forming linguistic variable, described by a triplet of the form C L = { 〈 c l i , U c l , C L ˜ 〉 } , c l i ∈ T ( u c l ) , ∀ i ∈ [ 0 , C a r d U C L ] , where T i ( u c l ) is extended term set of the linguistic variable “Closeness” from Table 1, C L ˜ is normal fuzzy set with correspondent membership function μ c l : U C L → [ 0 , 1 ] .

We will use the following mapping α : C L ˜ → U C L | u c l = E n t [ ( C a r d U C L − 1 ) × c l n o r m ] | ∀ i ∈ [ 0 , C a r d U C L ] , were

C L ˜ = ∫ U c l μ c l ( u c l ) / u c l (2.49)

On the other hand, similarly to the previous cases, to determine the estimates of the membership function in terms of singletons from (2.49) in the form μ c l i ( c l i ) / c l i | ∀ i ∈ [ 0 , C a r d U C L ] we propose the following procedure.

∀ i ∈ [ 0 , C a r d U C L ] , μ c l ( c l i ) = 1 − 1 C a r d U C L − 1 × | i − E n t [ ( C a r d U C L − 1 ) × c l n o r m ] | (2.50)

We also represent Δ x . . from (2.8), (2.15), (2.22), (2.29), (2.36), (2.43) and also Δ y . . from (2.10), (2.17), (2.24), (2.31), (2.38), (2.45) and finally Δ z . . from (2.12), (2.19), (2.26), (2.33), (2.40), (2.47) as a fuzzy set, forming linguistic variable, described by a triplet of the form

S T = { 〈 s t j , U S T , S T ˜ 〉 } , s t j ∈ T ( u s t ) , ∀ j ∈ [ 0 , C a r d U S T ] , where T j ( u s t ) is extended term set of the linguistic variable “Steps_scale” from Table 1, S T ˜ is normal fuzzy set with correspondent membership function μ s t : U s t [ 0 , 1 ] .

We will also use the following mapping Ω : S T ˜ → U S T | u s t = E n t [ ( C a r d U S T − 1 ) × s t j ] | ∀ j ∈ [ 0 , C a r d U S T ] , were

S T ˜ = ∫ U S T μ s t ( u s t ) / u s t . (2.51)

On the other hand, similarly to the previous cases, to determine the estimates of the membership function in terms of singletons from (2.51) in the form μ s t j ( s t j ) / s t j | ∀ j ∈ [ 0 , C a r d U S T ] we propose the following procedure.

∀ j ∈ [ 0 , C a r d U S T ] , μ s t j ( s t j ) = 1 − 1 C a r d U S T − 1 × | j − E n t [ ( C a r d U S T − 1 ) × s t j ] | (2.52)

To convert (2.49)-(2.52) into fuzzy logic-based statement and terms from Table 1 we use a Fuzzy Conditional Inference Rule, formulated by means of “common sense” as a following conditional clause:

P = “IF ( C L ˜ is CL), THEN ( S T ˜ is ST)” (2.53)

In other words, we use fuzzy conditional inference of the following type [3] [4] [5] [6] :

Ant 1: If Closeness is CL then Steps_scale is ST

Ant2: Closeness is CL’

--------------------------------------------------------------- (2.54)

Cons: Steps_scale is ST’.

Where C L , C L ′ ⊆ U C L and S T , S T ′ ⊆ U S T .

Note that statement (2.54) represents “modus-ponens” syllogism. Given that we use the following type of implication [7]

A → B = { ( 1 − a ) ⋅ b , a > b , 1 , a ≤ b (2.55)

Now for fuzzy sets (2.49) and (2.51) a binary relationship for the fuzzy conditional proposition for fuzzy logic with implication of type (2.55) is defined as

R ( A 1 ( c l ) , A 2 ( s t ) ) = C L × U C L → S T × U S T = ∫ U C L × U S T ( μ c l ( u c l ) / u c l → μ s t ( u s t ) / u s t ) / ( u c l , u s t ) (2.56)

and since we consider that C a r d U C L = C a r d U S T , then expression (2.56) looks like

μ c l ( u c l ) / u c l → μ s t ( u s t ) / u s t = { ( 1 − μ c l ( u c l ) ) ⋅ μ s t ( u s t ) , μ c l ( u c l ) > μ s t ( u s t ) , 1 , μ c l ( u c l ) ≤ μ s t ( u s t ) . (2.57)

By using (2.54) and given a unary relationship R ( A 1 ( c l ′ ) ) = C L ′ one can obtain the consequence R ( A 2 ( s t ′ ) ) by compositional rule of inference (CRI) to R ( A 1 ( c l ′ ) ) and R ( A 1 ( c l ) , A 2 ( s t ) ) of type (2.57):

R ( A 2 ( s t ′ ) ) = C L ′ ∘ R ( A 1 ( c l ) , A 2 ( s t ) ) = ∫ U C L μ c l ′ ( u c l ) / u c l ∘ ∫ U C L × U S T μ c l ( u c l ) → μ s t ( u s t ) / ( u c l , u s t ) = ∫ U S T ∪ c l ∈ U c l ( [ μ c l ′ ( u c l ) ∧ μ c l ( u c l ) ] → μ s t ( u s t ) ) / u s t . (2.58)

But for practical purposes we will use another Fuzzy Conditional Rule (FCR)

R ( A 1 ( c l ) , A 2 ( s t ) ) = ( C L × U C L → S T × U S T ) ∩ ( ¬ C L × U C L → U S T × ¬ S T ) = ∫ U C L × U S T ( μ c l ( u c l ) → μ s t ( u s t ) ) ∧ ( ( 1 − μ c l ( u c l ) ) → ( 1 − μ s t ( u s t ) ) ) / ( u c l , u s t ) (2.59)

Given (2.57) from (2.59) we are getting

R ( A 1 ( c l ) , A 2 ( s t ) ) = ( μ c l ( u c l ) → μ s t ( u s t ) ) ∧ ( ( 1 − μ c l ( u c l ) ) → ( 1 − μ s t ( u s t ) ) ) = { ( 1 − μ c l ( u c l ) ) ⋅ μ s t ( u s t ) , μ c l ( u c l ) > μ s t ( u s t ) , 1 , μ c l ( u c l ) = μ s t ( u s t ) , ( 1 − μ s t ( u s t ) ) ⋅ μ c l ( u c l ) , μ c l ( u c l ) < μ s t ( u s t ) . (2.60)

As it was mentioned above, “human like behavior” approach toward of an object control, namely “the further an Object from a Target, the faster an Object has to move (the larger step we have to make)” or in terms of linguistic variables from Table 1 “Closeness” and “Steps_scale” we use the following conditional clause:

P = “IF (CL is ‘smallest’), THEN (ST is ‘largest’)” (3.1)

To build a binary relationship matrix of type (2.53) and its basic realization (3.1) we use a conditional clause of type (2.60).

To build membership functions for fuzzy sets CL and ST we use (2.50) and (2.52) respectively.

In (2.50) the membership functions for fuzzy set CL (for instance from Table 1) would look like:

μ C L ( “ s m a l l e s t ” ) = 1 / 0 + 0.9 / 1 + 0.8 / 2 + 0.7 / 3 + 0.6 / 4 + 0.5 / 5   + 0.4 / 6 + 0.3 / 7 + 0.2 / 8 + 0.1 / 9 + 0 /   (3.2)

Note, that the membership function (2.52) for fuzzy set ST from Table 1 is

μ S T ( “ l a r g e s t ” ) = 0 / 0 + 0.1 / 1 + 0.2 / 2 + 0.3 / 3 + 0.4 / 4 + 0.5 / 5   + 0.6 / 6 + 0.7 / 7 + 0.8 / 8 + 0.9 / 9 + 1 / 10 (3.3)

Given (2.60), (3.2) and (3.3) we have R ( A 1 ( x ) , A 2 ( y ) ) shown in Table 2.

Suppose that the current value of “Closeness”, represented by a fuzzy set CL’ from (2.49), is defined as

μ C L ( “ l a r g e ” ) = 0.1 / 0 + 0.3 / 1 + 0.4 / 2 + 0.5 / 3 + 0.6 / 4 + 0.7 / 5   + 0.8 / 6 + 0.9 / 7 + 1 / 8 + 0.9 / 9 + 0.8 / 10

After applying CRI from (2.58), given an inference of a type (2.54) we get the following

R ( A 2 ( s t ) ) = 0.8 / 0 + 0.9 / 1 + 1 / 2 + 0.9 / 3 + 0.8 / 4 + 0.7 / 5   + 0.6 / 6 + 0.5 / 7 + 0.4 / 8 + 0.3 / 9 + 0.2 / 10

In other words, we are getting

μ S T ′ ( “ s m a l l ” ) = 0.8 / 0 + 0.9 / 1 + 1 / 2 + 0.9 / 3 + 0.8 / 4 + 0.7 / 5   + 0.6 / 6 + 0.5 / 7 + 0.4 / 8 + 0.3 / 9 + 0.2 / 10

which means the “closer” an Object to a Target, the “smaller” step is needed. This basic principle is the foundation for defining the value of a step an Object must make on each iteration.

In this study we presume that in order to approach a Target by an Object the latter must make multiple iterations. Every iteration characterized by certain nonlinear scale, which consists of multiple steps. The number of steps is defined by the value of CardU_ST. All steps are strictly correlated with a speed of an Object. The bigger the step, the higher the speed. The scale for each subsequent iteration is shorter by length than one of its predecessors. The farther an Object locates from a Target, the more iterations are needed to fulfil the goal (DAL, DAR, …). It is important to mention that the number of iterations would never be known in advance and will be defined by the algorithm, described down below.

Let us define the scale and closeness for each iteration k in X coordinate as the following

∀ k ∈ [ 0 , ⋯ ] ,   s c a l e k . = Δ x . . ,   c l o s k . = c l X (3.4)

where Δ x . . is defined in (2.8), (2.15), (2.22), (2.29), (2.36), (2.43).

Whereas clX is from (2.9), (2.16), (2.23), (2.30), (2.37), (2.44).

Define all j steps for each iteration k

∀ j ∈ [ 0 , C a r d U S T ] ;     s t e p k j = s c a l e k . / C a r d U S T ;     s c a l e k . = s c a l e k . − s t e p k j (3.5)

The procedure (3.5) would be resulted for each iteration k as the following nonlinear sequence

∀ k ∈ [ 0 , ⋯ ] ,     s t e p k . = s t e p k 0 , s t e p k 1 , s t e p k 2 , ⋯ , s t e p k C a r d U S T (3.6)

For instance, if C a r d U S T = 10 the following nonlinearity is taking place (in % of the size of an original s c a l e k . )

∀ k ∈ [ 0 , ⋯ ] ,     s t e p k . = [ 10.0 ; 9.0 ; 8.1 ; 7.29 ; 6.56 ; 5.9 ; 5.31 ; 4.78 ; 4.3 ; 3.87 ; 3.49 ]

Present c l o s k . ∈ [ c l o s min . , c l o s max . ] as a fuzzy set C L ′ ˜ of type (2.49) with correspondent membership function (2.50). Where c l n o r m = c l o s k . − c l o s min . c l o s max . − c l o s min . .

Applying CRI from (2.58), given an inference of a type (2.54) and Binary relationship matrix from Table 2, we get S T ′ ˜ of form (2.51).

Represent S T ′ ˜ as a sum of singletons

S T ′ ˜ = ∑ j = 0 C a r d U S T μ s t ( u s t j ) u s t j = μ s t ( u s t 0 ) / u s t 0 + μ s t ( u s t 1 ) / u s t 1 + ⋯ + μ s t ( u s t C a r d U S T ) / u s t C a r d U S T (3.7)

Since S T ˜ is a triangular normal membership function, we are having the following

∀ j ∈ [ 0 , C a r d U S T ] ;     ∃ ! j . * | μ s t ( u s t j . * ) = max { μ s t ( u s t j ) } = 1 (3.8)

Reduce the distance between an Object and a Target by value of a current step, associated with found j . * index.

Δ x . . = Δ x . . − s t e p k j . * (3.9)

Redefine coordinates of an Object (presume, that a Target is stationary)

x max O . . = x max O . . + Δ x . . ;     x min O . . = x min O . . + Δ x . . ;     x c e n t e r O . . = x c e n t e r O . . + Δ x . . (3.10)

Go to the next k + 1 iteration if a certain condition is met.

k = k + 1 | c l o s k . ≤ ε (3.11)

where ε is empirically defined threshold. The same algorithm (3.7)-(3.11) is applied to Y and Z coordinates by using correspondent

∀ k ∈ [ 0 , ⋯ ] ,     s c a l e k . = Δ y . . ,     c l o s k . = c l Y ,

∀ k ∈ [ 0 , ⋯ ] ,     s c a l e k . = Δ z ,     c l o s k . = c l Z .