DEVELOPING A METHOD FOR DETERMINING THE TIME PARAMETERS OF A MOBILE FIRE EXTINGUISHER OPERATOR

The object of this study is the process of functioning of the "man-machine" system on the example of the operator of a mobile fire installation. One of the issue when building models of such systems is to determine the parameters a priori of a given model of the human operator – the delay time and a time constant. For one of the promising means of fire extinguishing such as a mobile installation based on Segway, a method for determining the time parameters of the operator has been devised. A feature of the method is the use of approximation of partial derivatives from the phase-frequency characteristics of the operator in frequency, determined at two frequencies. This approach makes it possible to get rid of the need to use transcendental equations to determine time parameters and move on to an algebraic equation. To substantiate the values of frequencies at which partial derivatives are approximated, tolerance accuracy criteria are used. It is shown that working range of the operator of the mobile fire installation is in the infra-frequency region. Therefore, it is advisable to determine the phase-frequency characteristics of the operator numerically using an array of data on the transition function of the operator. An array of such data is formed using the Kotelnikov-Nyquist-Shannon theorem. A list of sequential procedures for the implementation of the method for determining the time parameters of the operator of a mobile fire installation is provided. The method for determining the time parameters of the operator of a mobile fire installation was verified by solving a test problem. It is shown that with permissible errors in the time parameters of the operator at the level of 5.0 %, the errors in their determining do not exceed 2.0 %. The reported results can be used for determining the dynamic parameters of the model of the operator of the fire installation, provided that the tolerance criterion for accuracy is set


Introduction
Among the means of fire extinguishing, an important place is occupied by vehicles. For example, the decisive role in extinguishing a fire in the Cathedral of Notre Dame (April 2019) was played by the mobile installation "Colossus" designed by Shark Robotics (France) [1]. In 2018, a fire drone was successfully tested in Latvia, which can extinguish fires at an altitude of 300 m [2]. A promising means of fire extinguishing is a mobile fire installation based on Segway [3]. Management of such a mobile fire installation both under a patrol mode and in the fire extinguishing mode is carried out by the operator. For the most efficient use of mobile fire installations, their characteristics must be coordinated with the characteristics of the operator [4]. In this regard, there is a need to build mathematical models of operators of such mobile fire installations or to identify the parameters of such models with their a priori known structure. In addition, when operating mobile fire installations, there is a need to test the operators of these systems. In this case, information on the parameters and characteristics of operators can be used. Such parameters include, in particular, time parameters -the time of delay and the time constant of the operator of the mobile fire installation. Thus, the task of determining the parameters a priori for a given mathematical model of the operator of a mobile fire installation is relevant.

V i t a l i y S o b y n a
PhD, Associate Professor, Head of Department*

D m i t r y S o k o l o v
PhD, Associate Professor*

S e r h i i R a h i m o v
PhD, Associate Professor* *Department of Organization and Technical Support of Emergency Rescue Works** **National University of Civil Defence of Ukraine Chernishevska str., 94, Kharkiv, Ukraine, 61023

The object of this study is the process of functioning of the "man-machine" system on the example of the operator of a mobile fire installation. One of the issue when building models of such systems is to determine the parameters a priori of a given model of the human operator -the delay time and a time constant.
For one of the promising means of fire extinguishing such as a mobile installation based on Segway, a method for determining the time parameters of the operator has been devised. A feature of the method is the use of approximation of partial derivatives from the phase-frequency characteristics of the operator in frequency, determined at two frequencies. This approach makes it possible to get rid of the need to use transcendental equations to determine time parameters and move on to an algebraic equation. To substantiate the values of frequencies at which partial derivatives are approximated, tolerance accuracy criteria are used.

It is shown that working range of the operator of the mobile fire installation is in the infra-frequency region. Therefore, it is advisable to determine the phase-frequency characteristics of the operator numerically using an array of data on the transition function of the operator. An array of such data is formed using the Kotelnikov-Nyquist-Shannon theorem. A list of sequential procedures for the implementation of the method for determining the time parameters of the operator of a mobile fire installation is provided. The method for determining the time parameters of the operator of a mobile fire installation was verified by solving a test problem. It is shown that with permissible errors in the time parameters of the operator at the level of 5.0 %, the errors in their determining do not exceed 2.0 %. The reported results can be used for determining the dynamic parameters of the model of the operator of the fire installation, provided that the tolerance criterion for accuracy is set
Keywords: mobile fire installation, operator, time parameters, dynamic parameters, frequency characteristics execution were obtained. However, this procedure does not make it possible to determine such individual parameters of the operator as the time of delay and the time constant. The author of work [17] studied the process of adaptation of the human operator to perform typical tasks of the control system. A time parameter such as the time of delay of the human operator was set a priori while the algorithm for its control was absent. It should be noted that the emergence of new technical solutions in the construction of automated control systems requires the creation of new models, or the correction of existing models of the human operator. In particular, this concerns the emergence of a new class of mobile fire installations on the basis of Segways [18], which are controlled by the operator. It should be emphasized that the dynamic properties of such an operator fully determine its time parameters -the delay time and the time constant.
All this gives reason to argue that it is expedient to conduct a study aimed at building a method for determining operator parameters, in particular, time parameters, which include the delay time and the operator's time constant.

The aim and objectives of the study
The aim of our work is to build a method for determining the time parameters of the operator of a mobile fire installation, which characterize its lateness and inertial properties. This will make it possible to coordinate the characteristics of the mobile fire installation with the capabilities of the human operator.
To accomplish the aim, the following tasks have been set: -to construct mathematical dependences for determining the time parameters of the operator; -to justify the choice of parameters for mathematical dependences that are used in determining the time parameters of the operator; -to compile a list of sequential procedures that enable determining the time parameters of the operator.

The study materials and methods
The object of this study is the process of functioning of the "man-machine" system on the example of a human operator and a mobile fire installation. The main hypothesis of the study assumes the possibility to determine the dynamic parameters of the model of the operator of a mobile fire installation using an array of data on the transition function of the operator. At the same time, to substantiate the values of frequencies at which partial derivatives are approximated, tolerance accuracy criteria are used.
The phase-frequency characteristics of the operator of a mobile fire installation are determined numerically on the basis of experimental data on its transient function. An array of experimental data is formed according to the Kotelnikov-Nyquist-Shannon theorem. Mathematical dependences for determining the time parameters of an operator are represented as a fourth-order algebraic equation. To find the roots of this algebraic equation, the Descartes-Euler, Ferrari methods, or the Maple application package (Canada) are used. The formation of initial data when substantiating mathematical dependences for determining the time parameters of an operator is carried out using mathematical analysis methods.

Literature review and problem statement
With regard to the mathematical description of human operator activity, it should be noted that there is a tendency to take into account various factors. So, for example, in [5] it is noted that the human operator plays a very important role, especially in the event of an emergency in industrial production. There is a need to work on such emergencies and related problems for the operator. However, there is no information in the cited work regarding models that reflect the behavior of the operator in such situations. This type of research also includes paper [6], which declares the need for "accurate determining of human behavior" when it is included in the automated system. This requires objective indicators that flow from the corresponding mathematical models. Such models are not given in the cited work. It is not clear what the author means when the term "reliable human-operator models" is used. The study does not provide an indicator of the reliability of the model. It should be noted that most of the developed models do not take into account such factors as experience, load, etc. [7], which is associated with the individual characteristics of the human operator. One of the ways out of this situation is to use integral indicators in human operator models. In particular, in [8] attention is paid to the fact that with an increase in the number of experiments there is a decrease in the values of the dynamic parameters of the operator of the fire installation. However, in the cited work there is no formalized description of this trend. The authors of [9] note that man and robot can perform only tasks that involve physical interaction. The concept of physical interaction between man and robot (pHRI) requires new approaches for its mathematical support and technical implementation. Thus, within the framework of this concept, work [10] reports the development of controllers, the use of which "softens" working conditions of a person when performing tasks of the corresponding class. The mathematical support of human activity in this case is reduced to a trivial level. Paper [11] reports the results of research on humanrobot interaction using appropriate gestures. The method of controlling a wheeled robot using human operator gestures is given in [12]. This approach has been worked out on the mobile robot E-puck (Switzerland). These works apply the same approach, which boils down to the implementation of appropriate algorithms that use information from primary sensors -accelerometers and gyroscopes. Mathematical models of the human operator in the traditional sense within the framework of the concept of human-robot interaction are not used. The authors of work [13] point to the need for constant monitoring of the state of the human operator, due to the maintenance at the appropriate level of the effectiveness of the functioning of the cyberphysical system. An approach to the formation of a human-operator model is provided, which is based on the concept of a hierarchical representation of the virtual model of a complex system and the synergistic method of basic functions. However, this approach determines the complexity of the human-operator model and the constant adjustment of its parameters. To determine the characteristics of the human operator, imitators of the technological process are used [14], as well as "man in the circuit" simulators [15]. As a rule, with this approach, the characteristics of the human operator are determined, which reflect his integral properties. In [16], based on the virtual reality methodology, estimates of such operator parameters as the response time to an emergency and the time of the task 5. Results of research on the development of a method for determining the time parameters of the operator of a mobile fire installation

1. Construction of mathematical dependences for determining the time parameters of the operator
The method for determining the time parameters of the operator of a mobile fire installation in the frequency domain is based on the use of its amplitude-frequency A(ω) and phase-frequency φ(ω) characteristics. These frequency characteristics are related through the complex transfer function W( jω) of the mobile fire installation operator, i. e.
where ω is the circular frequency; j -imaginary unit.
If the complex transfer function of the mobile fire installation operator is described by the expression where K -transmission coefficient; τ 0 , τ 1 -time parameters, then taking into account the dependence expression (3) is transformed as follows where ( ) ( ) ( ) This makes it possible to represent expressions (1) and (2) as (8), (9) are the basis for devising a method for determining the time parameters of a mobile fire installation operator in the frequency domain. These time parameters fully characterize the dynamic properties of the operator of the mobile fire installation.
According to (9), for derivatives at frequencies ω 1 and ω 2 , the following expressions hold ( ) ( ) As a result, for the time parameter τ 0 , two ratios can be written If we take into account that then, after combining expressions (12) and (13), the fourth-order algebraic equation with respect to the time parameter τ 1 is built where 2 1 ; To find the time parameter τ 1 , which is the root of algebraic equation (14), it is necessary to have information on the values of frequencies ω 1 , ω 2 , Δω and the values of the phase-frequency characteristic of the operator at frequencies ω 1 , ω 2 and ω 2 +Δω. Time parameter τ 0 is determined after finding the root of algebraic equation (14) using one of ratios (12) where expressions (13) are taken into account. To find the root of algebraic equation (14), the methods of Descartes-Euler, Ferrari, or the Maple application package can be used.

2. Justification of the choice of parameters for mathematical dependences for determining the time pa rameters of the operator
To determine the magnitude of the frequencies ω 1 and ω 2 , the ratio between the errors of the amplitude-frequency Δa and the phase-frequency Δφ characteristics of the operator and the errors of the time parameters Δτ 0 and Δτ 1 were used. These ratios are in the form ( ) where ΔA is the absolute error of the amplitude-frequency characteristic of the operator; a(ω)=A(ω)K -1 -the reduced amplitude-frequency characteristic of the operator. According to (16), the expression for the relative error δτ 1 of the time parameter τ 1 will be ( ) ( ) The relative error δτ 0 in accordance with (17) can be represented as follows ( ) ( ) After merging (18) and (19), the last expression is transformed to the form for which it is necessary to fulfill the condition The equivalent of this condition is a condition ( ) hence, the relationship between errors Δφ and Δa of the frequency characteristics of the operator The choice of the magnitude of the frequencies ω 1 and ω 2 is carried out according to the criteria 0 0 m a x ; δτ ≤ δτ  (18), (20), (21).
To determine the value of Δω, we use the expression for the relative methodical error δ, which is due to the approximation of derivatives according to expression (13). This error is described by the expression  Fig. 1 shows the dependence abs δ=f(ω, Δω), which is constructed for typical values of time parameters τ 0 =0.2 s and τ 1 =0.3 s, obtained in [8].

3. Development of a sequence of procedures that enable determining the time parameters of the operator
Determining the time parameters of the operator of a mobile fire installation is focused on the use of his frequency characteristics, as a result of which the question arises regarding their acquisition. At the same time, it should be noted that the operating frequency range for the operator of a mobile fire installation lies in the infra-low region and does not exceed (10.0÷12.0) s -1 .
Traditional methods for determining frequency characteristics are based on the use of sinusoidal signals, signals that approximate sinusoidal signals, or a field of harmonic signals. These methods have a number of disadvantages [19]: -the complexity of equipment for measurements in the infra-low-frequency range; -long measurement time; -the need for signal conversion; -measurement conditions and parameters of the object under study may change during measurements.
One of the options that is devoid of these shortcomings is the method of obtaining the frequency characteristics of the operator of a mobile fire installation, given in [20]. The method is based on the use of the transition function of the operator, which is determined experimentally at discrete moments of time with an interval Δt, which is formed according to the Kotelnikov-Nyquist-Shannon theorem.
In this case, the phase-frequency characteristic of the operator of the mobile fire installation is described by the expression  where Δ k is the increment of the transition function of the operator on the time interval Δt between the k+1-th and k-th measurements.
To do this, devices can be used, the scheme of one of which is given in [21].
The implementation of the method for determining the time parameters of the operator of a mobile fire installation involves performing the following sequence of procedures: -formation of requirements for the magnitudes of errors Δa and Δφ of the frequency characteristics of the operator (taking into account condition (26)); -determining, according to expression (22), the lower limit for the frequency ω 1 ; -formation of requirements for permissible values of relative errors δτ 0 max and δτ 1 max of the time parameters of the operator; -determining frequencies ω 1 and ω 2 according to conditions (24) and using expressions (18) and (20); -formation of requirements for the permissible value of the methodological error δ max ; -determining the value of the frequency Δω under condition (26) (using expression (25) or the nomogram shown in Fig. 2); -determining parameters A 1 , A 2 and B (according to expressions (13) and (15)) using data on the phase-frequency characteristic φ(ω) of the operator (according to expression (27)); -determining the value of the time parameter τ 1 as the root of algebraic equation (14), for example, using the Maple package; -determining the value of the time parameter τ 0 using ratios (12) and (13); -checking conditions (24). Example 2. For data in example 1, it is obtained that A 1 =-0,42 s, A 2 =-0,36 s. According to (15), B=0.06 s. Algebraic equation (14) takes the form 4 3 2 1 1 1 0.36 5.00 0.76 0.06 0, τ − τ + τ + = one of the roots of which is τ 1 =0.298 s. According to expressions (12) and (13), τ 0 =0.196 s. Errors in determining the time parameters τ 0 and τ 1 do not exceed 2.0 %.

Discussion of results of research on the development of a method for determining the time parameters of a mobile fire installation operator
The method for determining the time parameters of the operator of a mobile fire installation is based on the use of analytical dependences for his frequency characteristics. Unlike well-known methods, with the help of partial derivatives of the phase-frequency characteristic of the operator in frequency for two frequencies, a system of functions is built that includes time parameters. This system of functions acts as initial data in the formation of a mathematical justification of the method for determining the time parameters of the operator.
A feature of the activity of the operator of a mobile fire installation is that his operating frequency range lies in the infra-frequency region and does not exceed (10.0÷12.0) s -1 . This causes a number of technical difficulties in determining it, in particular, the phase-frequency characteristic. One of the features of the method is that when obtaining an expression for the phase-frequency characteristic of the operator (in the form of (27)), experimental data are used regarding the dynamic properties of the operator not in the frequency area but in the time domain. This makes it possible to simplify the procedure for determining the phase-frequency characteristics of the operator. Expression (27), obtained in this way, for the phase-frequency characteristic of the operator is used to construct an approximation (13) of the partial derivatives at two frequencies. The use of partial approximation of the phase-frequency function φ(ω) in frequency ω makes it possible to form analytical dependences to determine the time parameters of the operator. These dependences are in the form of a fourth-order algebraic equation (14) and relationships of the form (12). To use these dependences, it is necessary to have information regarding the magnitude of the frequencies ω 1 , ω 2 and Δω. These frequency parameters are determined using tolerance criteria for accuracy. A feature of this is that the absolute value of the error of the phase-frequency characteristic of the operator must be greater than the absolute value of the reduced error of the amplitude-frequency characteristic of the operator. The relationship between the errors of the operator's frequency characteristics and the errors of its time parameters makes it possible to build analytical relationships between the frequency parameters ω 1 and ω 2 and these errors (expressions (18) to (20)). To determine the value of the frequency parameter Δω, expression (25) is used, which describes the methodical error in the approximation of partial derivatives. In the operational determination of the value of the parameter Δω, the nomogram shown in Fig. 2 can be applied.
The method for determining the time parameters of the mobile fire installation operator was verified using the test problem solution, which showed that the errors in determining all time parameters do not exceed 2.0 %.
The advantage of the developed method is that due to the use of partial derivatives of the phase-frequency characteristics of the operator in frequency, the simplification of the equation used in determining the time parameters is provided. In particular, when using the expression directly for the phase-frequency characteristic of the operator, such an equation is transcendental, in contrast to the algebraic equation in the form (14).
The limitation of the developed method for determining the time parameters of the operator of a mobile fire installation is the minimum possible values of errors in the amplitude-frequency and phase-frequency characteristics of the operator.
The caveat of this method is the presence of individual characteristics of the human operator, which makes it difficult to coordinate his characteristics with the characteristics of a mobile fire installation.
Further development of the method may be associated with its application to other dynamic models of operators of fire installations.

Conclusions
1. Mathematical dependences have been built to determine the time parameters of the operator of a mobile fire installation, which are based on approximations of partial derivatives of his phase-frequency characteristic over frequency at two a priori given frequencies, which are determined using the metrological characteristics of the operator.
2. Using the tolerance criterion for accuracy, the choice of frequencies is justified, and it is shown that the lower limit of the values of these parameters is determined by the ratio between the errors of the phase-frequency and amplitude-frequency characteristics of the operator. Moreover, the lower limit of these frequencies is determined by the magnitude of the errors of the phase-frequency and amplitude-frequency characteristics of the operator, provided that the first exceeds the second.
3. A list of the sequence of procedures for determining the time parameters of the operator of a mobile fire installation has been developed, which involves the use as initial data of permissible values of relative errors for frequencies and for frequency characteristics of the operator. Using the solution to the test problem, the method for determining the time parameters of the operator of the mobile fire installation was verified and it was shown that the errors in determining them do not exceed 2.0 % with an allowable value of 5.0 %.

Conflicts of interests
The authors declare that they have no conflicts of interest in relation to the current study, including financial, personal, authorship, or any other, that could affect the study and the results reported in this paper.

Funding
The study was conducted without financial support.

Data availability
Manuscript has no associated data.