AIRCRAFT FUEL MEASUREMENT SYSTEM BASED ON HYDROSTATIC PRESSURE SENSORS

— The construction of an aircraft fuel-measuring system based on hydrostatic pressure sensors is considered, which makes it possible to determine the fuel residue in the aircraft tanks during its evolutions. With the evolution of aircraft, measuring the fuel residue in existing fuel metering systems using float and capacitive fuel level sensors has a rather complex electromechanical design and significant weight and size characteristics. This together affects the reliability of such systems as a whole and leads to significant methodological errors in determining the remaining fuel during maneuverable flight. The proposed system using hydrostatic pressure sensors and a computer can significantly increase the efficiency of existing fuel metering systems, and can also be used for calibration tests both on the ground and in flight.


I. INTRODUCTION
The errors of float and capacitance fuel gauges consist of the following components:  errors resulting from longitudinal and lateral rolls and accelerations of the airplane;  errors resulting from inaccurate installation of fuel tanks and deviations of their dimensions from those obtained during calculation and calibration;  temperature errors caused by changes in fuel temperature in the tank and change of fuel grade;  errors caused by changes in the voltage of the power supply.
The first three groups of errors are methodical errors.And if the errors associated with inaccurate installation of fuel tanks and temperature errors can be compensated by introducing additional calibration schemes, then to compensate for errors caused by the slope of the "fuel mirror" arising from changes in the angular orientation of the aircraft and the action of acceleration on the aircraft are developed not always successful algorithmic compensators.
The analysis of existing widespread fuel measuring systems (FMS) of aircraft using float and capacitive fuel level sensors shows that these systems measure fuel level in aircraft tanks with sufficient accuracy only in horizontal flight.During aircraft evolutions, the measurement of fuel residue in such FMS leads to significant methodological errors.In addition, they have a rather complex electromechanical design and significant mass-size characteristics, which together affects overall reliability of such vehicles as a whole.Therefore, studies aimed at the development of fuel measurement systems minimizing such methodological errors are very relevant.

II. PROBLEM STATEMENT
The known methods of measuring the fuel level in the tanks of modern aircraft using float and capacitive fuel level sensors do not allow to determine its residue during maneuvering, when pitch and roll angles undergo significant changes.In the conditions of maneuvering flight the methodical error increases significantly, which during prolonged maneuvering can lead to undesirable consequences.In order to obtain stable fuel residue readings for the above reasons, it is necessary to use new approaches to the construction of the measuring system of fuel residue in the tanks of the aircraft with high reliability, as well as with the minimum mass and dimensional dimensions.

III. PROBLEM SOLUTION
Let us consider the principle of FMS operation on the basis of hydrostatic pressure sensors (HPS) on the example of determining the volume of fuel in the tank of an aircraft made in the form of a parallelepiped with edges a, b, c with fuel "mirror height" h fm (Fig. 1).The base of the parallelepiped is tied to the horizontal geotopic triangle OENH, which in this case is considered as an instrumental one.The angles of heading Ψ, pitch ϑ and roll ϒ of the aircraft are assumed for this case to be zero.Modern designs of aircraft fuel tanks are usually located in the wings and fuselage of the airplane and have shapes close to a parallelepiped.To solve the problem of determining the fuel volume in each of the tanks, it is proposed to divide its space into inscribed volume figures, the fuel volume in which can be calculated separately, and the total fuel volume can be found as the sum of the fuel volumes of the figures inscribed in the tank.In this article, the tetrahedron (T) is considered as an inscribed volume figure .For example, let us consider the possibility of partitioning a parallelepiped into tetrahedrons.The parallelepiped, as can be seen from Modern designs of aircraft fuel tanks are usually located in the wings and fuselage of the airplane and a parallelepiped.To solve the problem of determining the fuel volume in each of the tanks, it is proposed to divide its space into inscribed volume figures, the fuel volume in which can be calculated separately, and the total fuel the sum of the fuel volumes of the figures inscribed in the tank.In this article, the tetrahedron (T) is considered as an inscribed volume For example, let us consider the possibility of parallelepiped into tetrahedrons.The parallelepiped, as can be seen from Fig. 2, is 1, with vertices: 1, 3, 4, 8; 2, with vertices: 1, 2, 3, 6; 3, with vertices: 3, 6, 7, 8; 4, with vertices: 1, 5, 6, 8. Fuel residue volume V x for tetrahedron No.2 with vertices 1, 2, 3, 6 (Fig. 3) in the coordinate system of the instrument trihedron OENH one of the base vertices can be determined fr following relations: 1) Total volume of the tank tetrahedron with vertices 1, 2, perpendicular side planes of the tetrahedron): where a, b is lengths of faces T; height of T according to the information from at vertex 2.
2) The volume of the unfilled part of the tank with vertices in points 1', 2', 3', 6 is determined by the lengths of the segments of the edges of the unfilled part of the tetrahedron the FM.The sought segments from the relations for rectangular similar triangles.Through similar triangles formed by vertices 1, 2, 6 and 1', 2', 6' the segment a triangles with vertices 3, 2, 6 and 3', 2', 6' the segment b' is determined: 3) The desired fuel volume is defined as the difference between V 0 and the volume of the unfilled part of the tetrahedron V up : To find the fuel residue in a tetrahedron of arbitrary shape, to calculate the lengths of the segments obtained from the intersection of the fm with the edges of the tetrahedron, it is necessary to recalculate the parameters of the vector perpendicular to the FM plane inclined edges.Such a problem of computing the volume V x in segments is solvable, but requires the use of a rather complex algorithm [1].

Electronics and Control Systems 2023. N 3(77): 77-82
Partitioning a parallelepiped into tetrahedrons for tetrahedron No.2 with 6 (Fig. 3) in the coordinate system of OENH and HPS located in one of the base vertices can be determined from the Total volume of the tank V 0 (volume of the 2, 3, 6 and two mutually perpendicular side planes of the tetrahedron): is lengths of faces T; h 0 is the calculated height of T according to the information from HPS The volume of the unfilled part of the tank V up with vertices in points 1', 2', 3', 6 is determined by the lengths of the segments of the edges of the tetrahedron a' and b' cut off by the FM.The sought segments a' and b' are found from the relations for rectangular similar triangles.Through similar triangles formed by vertices 1, 2, 6 a' is found, and through vertices 3, 2, 6 and 3', 2', 6' the The desired fuel volume is defined as the and the volume of the unfilled To find the fuel residue in a tetrahedron of arbitrary shape, to calculate the lengths of the segments obtained from the intersection of the fm with the edges of the tetrahedron, it is necessary to recalculate the parameters of the vector the FM plane h fm {0, h fm , 0} on its inclined edges.Such a problem of computing the in segments is solvable, but requires the plex algorithm [1].In general, the equation of the plane, linear with respect to Cartesian rectangular coordinates is [1]: where A, B and C not equal to zero simultaneously define the plane.
The equations of three side planes of a tetrahedron of arbitrary shape with one HPSS in one of the vertices of the base, which coincides with the horizontal plane of the instrumental tetrahedron OENH, can be represented in the following form:  plane with vertices 1, 6, 3  plane with vertices 1, 6, 2  plane with vertices 2, 6, 3 The parameters of the planes with respect to the OENH coordinate system are assumed to be known.
The equation of the FM plane passing through one of its points and perpendicular to the vector N{A, B, C} can be represented in the following form In the case of parallelism of the FM plane to the OENH plane (A = 0, C = 0), equation (6) will take the following form: for all points of the FM.The x and y coordinates of the FM for points 1', 2', 3' (Fig. 3) are found as the intersection of the tetrahedron planes (3) … (5) with the FM plane (7).
For the coordinates of points 1', 2', 3', respectively, we have three systems of equations and their solution with respect to the three points of intersection of the FM with the tetrahedron:  for coordinates of point 1'( 1ENH x , h fm 1', 1ENH z ): According to the found coordinates of FM and the known coordinates of the tetrahedron vertex V ENH x , V ENH y , V ENH z the volume of the unfilled part of fuel in the tank is determined by the formula The volume of the remaining fuel is found as the difference between the a priori known volume of the tetrahedron V 0 and the calculated unfilled part The obtained solution of the fuel determination problem was considered in the OENH system.At the same time, in the conditions of a fixed base and during the aircraft evolutions, the position of the instrument triangle OXYZ in accordance with the changes in its angular position relative to the triangle OENH changes in the angles of heading , pitch  of the aircraft.It follows that for finding the coordinates 1', 2', 3' of the FM intersection with the edges of the tetrahedron it is necessary to take into account the inclination of the fm plane relative to the movable instrumental trihedral OXYZ The relationship between the coordinates of the OENH triangles in OXYZ at the current moment of time can be found through the rectangular matrix of directional cosines in the form sin sin sin cos cos cos cos sin cos sin cos sin sin cos cos cos sin cos sin cos cos sin cos sin sin cos sin sin sin

B ISSN 1990-5548 Electronics and Control Systems 202
the volume of the unfilled part of fuel in the tank is determined by the formula [2]: The volume of the remaining fuel is found as the difference between the a priori known volume of the and the calculated unfilled part V up .The obtained solution of the fuel determination OENH coordinate system.At the same time, in the conditions of a fixed base and during the aircraft evolutions, the OXYZ will change ance with the changes in its angular OENH, i.e., with the , pitch ϑ, and roll of the aircraft.It follows that for finding the coordinates 1', 2', 3' of the FM intersection with the s of the tetrahedron it is necessary to take into account the inclination of the fm plane relative to the OXYZ (Fig. 4).

Tilt of the "fuel mirror" plane relative to the movable instrumental trihedral
The relationship between the coordinates of the at the current moment of time can be found through the rectangular matrix of sin sin sin cos cos sin cos sin cos sin sin cos cos cos sin cos sin cos cos sin sin cos sin sin sin .
In this case, the coordinates of the tetrahedron planes in the axes of the tetrahedron have the form

B
The angles , ϑ, and  obtained from the inertial navigation system (INS), the heading and vertical system and other meters available on board.
In these cases, the FM plane at points 1',2',3' in the case of its inclination relative to the moving instrument triangle OXYZ vector perpendicular to the FM plane it was at zero heading angles aircraft, but by three vectors {0, zero heading angles , ϑ and three: {0, hfm1', 0}, {0, hfm2', 0}, {0, hfm3 Thus, to calculate the heights h fm 3', it is necessary to have information from three HPS located at the vertices of the base of the tetrahedron (see Fig. 4).
Thus, to solve the problem of determ fuel residue in the mobile coordinate system at the current moment of time, it is necessary, in addition to the calculated values of h have information about the current coordinates of the vertices of the tetrahedron coefficients of its planes , , , 1 4 A B C D i the coordinate system OXYZ the current moment of time can be determined on the basis of the inverse matrix of directional cosines, obtained from the direct square matrix known angles , ϑ, : Based on relations (14) values of heights h fm 1', h fm 2' readings of three FMSs, the system of equations for In this case, the coordinates of the tetrahedron planes in the axes of the tetrahedron OENH will .
 of the aircraft can be obtained from the inertial navigation system (INS), the heading and vertical system and other meters In these cases, the FM plane at points 1',2',3' in the case of its inclination relative to the moving is defined not by one vector perpendicular to the FM plane N{0, h fm ,0}, as it was at zero heading angles , ϑ, and  of the aircraft, but by three vectors {0, h fm 1',0} as it was at ϑ and  of the aircraft, but {0, hfm1', 0}, {0, hfm2', 0}, {0, hfm3´, 0}.
Thus, to calculate the heights h fm 1', h fm 2' and 3', it is necessary to have information from three located at the vertices of the base of the Thus, to solve the problem of determining the fuel residue in the mobile coordinate system at the current moment of time, it is necessary, in addition h fm 1', h fm 2' and h fm 3', to have information about the current coordinates of the vertices of the tetrahedron x i , y i , z i and the , , , 1 4 These parameters at the current moment of time can be determined on the basis of the inverse matrix of directional cosines, obtained from the direct square matrix B through the Based on relations ( 14) -( 16) and calculated ' , h fm 3' according to the readings of three FMSs, the system of equations for  Based on relations (12) and ( 14)…( 19), the structure of the algorithm for determining the fuel residue at the current moment of time for a tank in the form of an arbitrary tetrahedron for a mobile aircraft can be represented as follows (Fig. 5).When realizing the method of measuring the fuel residue in the tanks of aircraft with the division of tank volumes into tetrahedrons of arbitrary shape, there is a methodological error associated with the oscillations of the fuel surface at the occurrence of aircraft acceleration.For estimation of sensor measurements in the selected time interval of pressure measurement it is supposed to use methods of optimal measurement processing.
At mismatch of planes of tetrahedrons inscribed in the tank volume with real planes of tanks for compensation of methodical error the alignment is required.The alignment procedure can be realized by measuring by means of FMS the readings of current fuel consumption at change of heading angles , ϑ and  of the aircraft and comparing them with the real readings of fuel residue in the tank.The difference in readings can be taken into account in flight in the form of corrections coming from the calculator.

IV. CONCLUSIONS
The use of hydrostatic pressure sensors in the system of fuel residue measurement and calculator allows to solve the problem of increasing the accuracy of the aircraft fuel measuring system in all flight modes and to reduce its mass and dimensional characteristics.

Fig. 1 .
Fig. 1.Determining the volume of fuel in a parallelepiped-shaped tankAccording to the readings of the HPS, which is located in one of the vertices of the base of the parallelepiped, we can calculate the height of the "fuel mirror" (FM) plane hfm.According to Pascal's law Fig. partitioned into four tetrahedrons:  tetrahedron 1, with vertices: 1,  tetrahedron 2, with vertices: 1,  tetrahedron 3, with vertices: 3,  tetrahedron 4, with vertices: 1, ISSN 1990-5548 Electronics and Control Systems 202 Determining the volume of fuel in a shaped tankAccording to the readings of the HPS, which is located in one of the vertices of the base of the parallelepiped, we can calculate the height of the "fuel mirror" (FM) plane hfm.According to Pascal's law

Fig. 3 .
Fig. 3. Volume of fuel residue for the tetrahedron One of the variants of another solution of the problem of fuel residue determination for an arbitrary tetrahedron can be realized with the transition from volume calculation in segments to volume calculation by coordinates of fm intersection with tetrahedron edges.In general, the equation of the plane, linear with respect to Cartesian rectangular coordinates is[1]:

Fig. 4 .
Fig. 4. Tilt of the "fuel mirror" plane relative to the movable instrumental trihedral

Fig. 5 .
Fig. 5. Structure of the algorithm for determining the fuel residue at the current moment of time