Adaptive Filtration of Parameters of the UAV Movement Based on the TDOA-Measurement Sensor Networks

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INTRODUCTION
Unmanned aerial vehicles (UAVs) are beginning to play an increasingly important role.The main arguments for their dissemination are low cost, high mobility and the ability to transfer photo-video information in real time.On the other hand, the availability and massive use of UAVs has led to a new class of threats (Nonami et al. 2010;Wallace and Loffi 2015;Sathyamoorthy 2015;Solodov et al. 2018;Friese et al. 2016;Card 2018;Dronebouncer 2018).
The development of wireless technology and microelectronics has led to the emergence and widespread use of wireless sensor networks (WSN), which are distributed networks self-organizing from miniature sensors that exchange information through wireless technology.One of the main uses of WSN is the creation of various monitoring and tracking systems of moving radio sources, including UAVs (Gemayel et al. 2014;Wan et al. 2016).
To determine the UAV location using wireless sensor networks, TDOA measurements are widely used (Rullan-Lara et al. 2013;Makki et al. 2016).At TDOA measurement, synchronization between WSN and a radio source is not necessary, which allows its use for positioning the unknown UAV.
Adaptive Filtration of Parameters of the UAV Movement Based on the TDOA-Measurement Sensor Networks xx/xx 02/10 The UAV trajectory as a rule consists of non-overlapping sections with different types of movement.It can be presented in the form of stochastic process from which probability characteristics change at random instants of time.A convenient mathematical model of such processes are stochastic discrete dynamic systems with a random structure, which are adequate to solvable tasks when realization of algorithms on digital computers.To increase the accuracy of the UAV coordinates estimate on sections with different types of movement, various measurement processing algorithms must be used.However, the type of movement is, as a rule, unknown.Therefore, besides estimating the UAV coordinates, it is also necessary to solve the task of recognizing its motion type.
In the work of Tovkach and Zhuk (2017a) the adaptive algorithm is synthesized, with estimation of the UAV movement parameters and its movement type.Observations at synthesis of the algorithm are the UAV location coordinates in the rectangular coordinate system, which are determined on the basis of TDOA measurements using the least squares method.The disadvantage of the obtained algorithm is that definition the UAV location is carried out after arrival of measurements from all sensors, which leads to the necessity inversion of large-dimensional matrices.
Therefore, the synthesis of adaptive algorithms for the UAV movement parameters estimate based on the TDOA-measurement sensor networks, which provide serial processing of measurements, is important and do not require the inversion of largedimensional matrices.

PROBLEM DEFINITION
The UAV movement with various types of maneuver in rectangular system of coordinates can be described by a stochastic dynamic system with random structure (Eq. 1) (Zhuk 1989): The UAV movement with various types of maneuver in rectangular system of coordinates can be described by a stochastic dynamic system with random structure (Eq. 1) (Zhuk 1989): ( ) ( 1) ( ), 1, , tr tr tr tr tr j j where: u tr (k) is the state vector; tr j F and tr j G are the matrixes describing various types of movement; ω tr (k) is the uncorrelated sequence of Gaussian vectors with a single correlation matrix.
During propagation of a signal from the UAV to sensors of network, its coordinates do not change.Therefore, it is assumed that the measurements of the differences of distances between the sensors and the reference sensor are received in the k-th moment of time.The moment of receipt of a signal to the basic sensor with coordinates of x 0 , y 0 , z 0 is used as the k-th moment of time.
Distance difference measurement errors are correlated because they contain reference sensor measurement error.The presence of correlated errors makes it difficult to use traditional recurrent target coordinates evaluation algorithms.This difficulty can be avoided through the introduction of error υ 0 (k) into the state vector of the estimated parameters (Tovkach and Zhuk 2017b), which can be represented in the form u T (k) = (u trT (k), υ 0 (k)).In this case, the model of the process to be filtered is transformed to Eq. 2 ( ) ( 1) ( ), 1, , where: ω T (k) = (ω trT (k), ωΔ(k)); ωΔ(k) is the uncorrelated sequence of Gaussian values with a single tr tr

 
The UAV movement with various types of maneuver in rectangular system of coordinates can be described by a stochastic dynamic system with random structure (Eq. 1) (Zhuk 1989): ( ) ( 1) ( ), 1, , tr tr tr tr tr j j where: u tr (k) is the state vector; tr j F and tr j G are the matrixes describing various types of movement; ω tr (k) is the uncorrelated sequence of Gaussian vectors with a single correlation matrix.
During propagation of a signal from the UAV to sensors of network, its coordinates do not change.Therefore, it is assumed that the measurements of the differences of distances between the sensors and the reference sensor are received in the k-th moment of time.The moment of receipt of a signal to the basic sensor with coordinates of x 0 , y 0 , z 0 is used as the k-th moment of time.
Distance difference measurement errors are correlated because they contain reference sensor measurement error.The presence of correlated errors makes it difficult to use traditional recurrent target coordinates evaluation algorithms.This difficulty can be avoided through the introduction of error υ 0 (k) into the state vector of the estimated parameters (Tovkach and Zhuk 2017b), which can be represented in the form u T (k) = (u trT (k), υ 0 (k)).In this case, the model of the process to be filtered is transformed to Eq. 2 ( ) ( 1) ( ), 1, , where: ω T (k) = (ω trT (k), ωΔ(k)); ωΔ(k) is the uncorrelated sequence of Gaussian values with a single dispersion; 0 0 0 .
In this case, the observation equation that describes the process of measuring the coordinates of UAV by sensor network is Eq. 3 The UAV movement with various types of maneuver in rectangular system of coo described by a stochastic dynamic system with random structure (Eq. 1) (Zhuk 1989): where: u tr (k) is the state vector; During propagation of a signal from the UAV to sensors of network, its coor change.Therefore, it is assumed that the measurements of the differences of distanc sensors and the reference sensor are received in the k-th moment of time.The moment signal to the basic sensor with coordinates of x 0 , y 0 , z 0 is used as the k-th moment of time.
Distance difference measurement errors are correlated because they contain re measurement error.The presence of correlated errors makes it difficult to use traditional coordinates evaluation algorithms.This difficulty can be avoided through the introductio into the state vector of the estimated parameters (Tovkach and Zhuk 2017b), which can be the form u T (k) = (u trT (k), υ 0 (k)).In this case, the model of the process to be filtered is transf where: .
In this case, the observation equation that describes the process of measuring the UAV by sensor network is Eq. 3 The UAV movement with various types of maneuver in rectangular system of coordinates can be described by a stochastic dynamic system with random structure (Eq. 1) (Zhuk 1989): where: u tr (k) is the state vector; tr j F and tr j G are the matrixes describing various types of movement; ω tr (k) is the uncorrelated sequence of Gaussian vectors with a single correlation matrix.
During propagation of a signal from the UAV to sensors of network, its coordinates do not change.Therefore, it is assumed that the measurements of the differences of distances between the sensors and the reference sensor are received in the k-th moment of time.The moment of receipt of a signal to the basic sensor with coordinates of x 0 , y 0 , z 0 is used as the k-th moment of time.
Distance difference measurement errors are correlated because they contain reference sensor measurement error.The presence of correlated errors makes it difficult to use traditional recurrent target coordinates evaluation algorithms.This difficulty can be avoided through the introduction of error υ 0 (k) into the state vector of the estimated parameters (Tovkach and Zhuk 2017b), which can be represented in the form u T (k) = (u trT (k), υ 0 (k)).In this case, the model of the process to be filtered is transformed to Eq. 2 where: .
In this case, the observation equation that describes the process of measuring the coordinates of UAV by sensor network is Eq. 3 (2) (3) where u tr (k) is the state vector; F j tr and G j tr are the matrixes describing various types of movement; ω tr (k) is the uncorrelated sequence of Gaussian vectors with a single correlation matrix.
During propagation of a signal from the UAV to sensors of network, its coordinates do not change.Therefore, it is assumed that the measurements of the differences of distances between the sensors and the reference sensor are received in the k-th moment of time.The moment of receipt of a signal to the basic sensor with coordinates of x 0 , y 0 , z 0 is used as the k-th moment of time.
Distance difference measurement errors are correlated because they contain reference sensor measurement error.The presence of correlated errors makes it difficult to use traditional recurrent target coordinates evaluation algorithms.This difficulty can be avoided through the introduction of error υ 0 (k) into the state vector of the estimated parameters (Tovkach and Zhuk 2017b), which can be represented in the form u T (k) = (u trT (k), υ 0 (k)).In this case, the model of the process to be filtered is transformed to Eq. 2: where ω T (k) = (ω trT (k), ω Δ (k)); ω Δ (k) is the uncorrelated sequence of Gaussian values with a single dispersion; In this case, the observation equation that describes the process of measuring the coordinates of UAV by sensor network is Eq.3: where Δ s (k) is the measured difference of distances between the s-th sensor and the reference sensor in the k-th moment of time: xx/xx 03/10 s = 1,S; υ s (k) is the measurement error of s-th sensor with dispersion d s ; h s (u) is the nonlinear function, described by Eq. 4: (4) s (k) is the measured difference of distances between the s-th sensor and the reference sensor in oment of time, 1, s S = ; υ s (k) is the measurement error of s-th sensor with dispersion d s ; h s (u) is near function, described by Eq. 4 (k), y(k), z(k) are coordinates of the UAV position; x s , y s , z s are coordinates of the s-th sensor υ 0 (k) is the error of the distances measurement by the reference sensor with dispersion d 0 .
ments obtained from all sensors are denoted as a vector uΔ(k n the basis of Eqs. 2 and 3 it is necessary to synthesize optimal and quasi-optimal adaptive algorithms of UAV movement parameters according to sensor network, in which the detection s types of motion is carried out.

DEVELOPMENT
quation 2 describes the process to be estimated, and Eq. 3 the formation process of data e to observation.For a description of the type of maneuver there is used the random variable pplying the technique of a condition vector expansion (Zhuk 1988) it is possible to show that the process (u(k), aj(k)) belongs to the class of the mixed Markov processes in discrete time with probability density (PD) (Eq.5) is the conditional PD defined by Eq. 2.
where: Δ s (k) is the measured difference of distances between the s-th sensor and the the k-th moment of time, Applying the technique of a condition vector expansion (Zhuk 1988) it is poss expanded process (u(k), aj(k)) belongs to the class of the mixed Markov processes i transition probability density (PD) (Eq.5) ( ( ), ( ) | ( 1), ( )) where: Π(u(k) | u(k -1), aj(k)) is the conditional PD defined by Eq. 2.
position; υ (k) is the error of the distances measurement by the reference sensor wit Measurements obtained from all sensors are denoted as a vector uΔ(k) = (Δ 1 (k),…,Δ S (k)) T .
On the basis of Eqs. 2 and 3 it is necessary to synthesize optimal and quasi-o filtration algorithms of UAV movement parameters according to sensor network, in whi of various types of motion is carried out.

DEVELOPMENT
Equation 2 describes the process to be estimated, and Eq. 3 the formation accessible to observation.For a description of the type of maneuver there is used the r Applying the technique of a condition vector expansion (Zhuk 1988) it is possible expanded process (u(k), aj(k)) belongs to the class of the mixed Markov processes in di transition probability density (PD) (Eq.5) ( ( ), ( ) | ( 1), ( )) where: Π(u(k) | u(k -1), aj(k)) is the conditional PD defined by Eq. 2.
On the basis of Eqs. 2 and 3 it is necessary to synt filtration algorithms of UAV movement parameters accordin of various types of motion is carried out.

DEVELOPMENT
Equation 2 describes the process to be estimated, accessible to observation.For a description of the type of m Applying the technique of a condition vector expansio expanded process (u(k), aj(k)) belongs to the class of the mix transition probability density (PD) (Eq.5) ( ( ), ( ) | ( 1), ( )) where: Π(u(k) | u(k -1), aj(k)) is the conditional PD defined b where x(k), y(k), z(k) are coordinates of the UAV position; x s , y s , z s are coordinates of the s-th sensor position; υ 0 (k) is the error of the distances measurement by the reference sensor with dispersion d 0 .Measurements obtained from all sensors are denoted as a vector u Δ (k) = (Δ 1 (k),…,Δ S (k)) T .On the basis of Eqs. 2 and 3 it is necessary to synthesize optimal and quasi-optimal adaptive filtration algorithms of UAV movement parameters according to sensor network, in which the detection of various types of motion is carried out.

DEVELOPMENT
Equation 2 describes the process to be estimated, and Eq. 3 the formation process of data accessible to observation.For a description of the type of maneuver there is used the random variable a j (k), j = 1,M which belongs to chains of Markov, accepting the M values with a matrix of probabilities of transitions Π ij (k, k -1) i, j = 1,M and initial probabilities p i (0), i = 1,M.Measurement errors υ 5 (k), s = 1,S are independent.
Applying the technique of a condition vector expansion (Zhuk 1988) it is possible to show that the expanded process (u(k), a j (k)) belongs to the class of the mixed Markov processes in discrete time with transition probability density (PD) (Eq.5): where Π(u(k) | u(k -1), a j (k)) is the conditional PD defined by Eq. 2.
The problem of synthesis of an optimal algorithm comes down to calculation of aposteriori PD of expanded process W(u(k), a j (k)) = P(u(k), a j (k) | U Δ (k)), where U Δ (k) = u Δ (1),…,u Δ (k) are the sequences of measurements received to till k-th moment, included.Introducing also designation of the extrapolated PD P(u(k), a j (k) | U Δ (k -1)) for the expanded process and following the technique of synthesis (Zhuk 1989), the optimal algorithm of a filtration can be presented in the form of two the recurrent Eqs.6 and 7: he problem of synthesis of an optimal algorithm comes down to calculation of aposteriori PD of ( ( 1), ( 1)) ( 1);  is one-step-likelihood function, defined on the basis of ents ( ), 1, The problem of synthesis of an optimal algorithm comes down to calculation of aposteriori PD of ( ( 1), ( 1)) ( 1); where:  is one-step-likelihood function, defined on the basis of measurements ( ), 1, and is also an optimal The problem of synthesis of an optimal algorithm comes down to calculation of aposteriori PD of …,uΔ(k) are the sequences of measurements received to till k-th moment, included.Introducing also designation of the extrapolated PD )) for the expanded process and following the technique of synthesis (Zhuk 1989), the optimal algorithm of a filtration can be presented in the form of two the recurrent Eqs.6 and 7 * 1 ( ( ), ( )) ( , 1) ( ( 1), ( 1)) ( 1); where:  is one-step-likelihood function, defined on the basis of measurements ( ), 1, and is also an optimal algorithm of extrapolation of mixed Markov processes (u(k), aj(k)) for one step.With the help of Eq. 7 the problem of synthesis of an optimal algorithm comes down to calculation of aposteriori PD of ( ( 1), ( 1)) ( 1); and is also an optimal f extrapolation of mixed Markov processes (u(k), aj(k)) for one step.With the help of Eq. 7 the The problem of synthesis of an optimal algorithm comes down to calculation of aposteriori PD of …,uΔ(k) are the sequences of measurements received to till k-th moment, included.Introducing also designation of the extrapolated PD )) for the expanded process and following the technique of synthesis (Zhuk 1989), the optimal algorithm of a filtration can be presented in the form of two the recurrent Eqs.6 and 7 * 1 ( ( ), ( )) ( , 1) ( ( ) | ( 1), ( )) ( ( 1), ( 1)) ( 1); where:  ( ( 1), ( 1)) ( 1); where:  ( ( 1), ( 1)) ( 1); where:  is one-step-likelihood function, defined on the basi measurements ( ), 1, The problem of synthesis of an optimal algorithm comes down to ca measurements received to till k-th moment, included.Introducing also desig P(u(k), aj(k) | UΔ(k -1)) for the expanded process and following the techniq the optimal algorithm of a filtration can be presented in the form of two the r * 1 ( ( ), ( )) ( , 1) ( ( ) | ( 1), ( )) ( ( 1), ( 1)) ( 1); where: where is one-step-likelihood function, defined on the basis of measurements is the conditional PD, defined by Eq. 6; P(u Δ (k) | U Δ (k -1)) is the conditional PD determined by: xx/xx 04/10 Equation 6 describes evolution of the extrapolated PD W * (u(k), a j (k)) and is also an optimal algorithm of extrapolation of mixed Markov processes (u(k), a j (k)) for one step.With the help of Eq. 7 the clarifying of extrapolated PD based on the obtained measurement u Δ (k) is made and a posteriori PD W(u(k), a j (k)) is defined.
The initial conditions for the algorithm Eqs. 8 to 11 have the form Wi(0) = pi(0 Using theEqs.8 and 11, extrapolated W * j(k) and a posteriori Wj(k) probabilities are c Eqs. 9 and 10 are conditional extrapolated W * j(u(k)) and a posteriori Wj(k) PD.The optimal a (Eqs.8 to 11) contains M channels and belongs to the class of devices with feedbacks betw (Zhuk 1988;1989).
) are the conditional extrapolated and a posteriori PD of the vector u(k) on condition a j (k); W * j (k) = P(a j (k), U Δ (k -1)) and W j (k) = P(a j (k), U Δ (k)) are extrapolated and a posteriori probabilities a j (k); P(u(k) | a j (k), U Δ (k -1)) is the conditional PD determined by: determined by: The initial conditions for the algorithm Eqs. 8 to 11 have the form W i (0) = p i (0), W i (u(0)) = P(u(0)), i = 1,M.Using the Eqs.8 and 11, extrapolated W * j (k) and a posteriori W j (k) probabilities are calculated and Eqs. 9 and 10 are conditional extrapolated W * j (u(k)) and a posteriori W j (k) PD.The optimal adaptive filter (Eqs.8 to 11) contains M channels and belongs to the class of devices with feedbacks between channels (Zhuk 1988;1989).Large computational costs are required for accomplishing the optimal algorithm.When synthesizing a quasi-optimal algorithm, there will be used a sequential method of processing incoming data.There will enter the index l, which characterizes the sequence of receipts of measurements from the corresponding sensors of network.The quasi-optimal algorithm of an adaptive filtration can be received by decomposition of vector-valued function (Eq.3), in a Taylor series in the vicinity of the points u(k) and restriction by linear members of decomposition, and also application on each step of Gaussian approximation of the conditional extrapolated PD W * j (u(k)).In this case the equation of calculation of the conditional extrapolated PD W * j (u(k)) (Eq.9) comes down to calculation of its first u * j (k) and second P * j (k) moments by Eqs. 12 and 13 (Zhuk 1989): Large computational costs are required for accomplishing the optimal algorithm.When synthesizing a quasi-optimal algorithm, there will be used a sequential method of processing incoming data.There will enter the index l, which characterizes the sequence of receipts of measurements from the corresponding sensors of network.The quasi-optimal algorithm of an adaptive filtration can be received by decomposition of vector-valued function (Eq.3), in a Taylor series in the vicinity of the points u(k) and restriction by linear members of decomposition, and also application on each step of Gaussian approximation of the conditional extrapolated PD W * j(u(k)).In this case the equation of calculation of the conditional extrapolated PD W * j(u(k)) (Eq.9) comes down to calculation of its first u * j(k) and second P * j(k) moments by Eqs. 12 and 13 (Zhuk 1989 xx/xx 05/10 Calculation of conditional a posteriori PD W j (u(k)) (Eq.10) at sequentially processing of the arriving measurements Δ l (k), l = 1,S comes down to calculation of its first û j (k) and second P ˆj(k) moments using the recurrent procedure (Eqs.14 to 16) (Evlanov and Zhuk 1990): j j P * j(k) moments by Eqs. 12 and 13 (Zhuk 1989) ( 1) / ( ) Equation of calculation of conditional a posteriori PD Wj(u(k)) (Eq.10) at sequentially processing of the arriving measurements ( ), 1, l k l S  = comes down to calculation of its first ûj(k) and second ˆ( ) j P k moments using the recurrent procedure (Eqs.14 to 16) (Evlanov and Zhuk 1990) ) u k P k are the expected value and correlation matrix of conditional a posteriori PD Wj(u(k)), refined by measurements ( ), 1, s k s l  = .In this case the initial conditions for the procedure Eqs.
( 1) / ( ) Equation of calculation of conditional a posteriori PD Wj(u(k)) (Eq.10) at sequentially pr of the arriving measurements ( ), 1, l k l S  = comes down to calculation of its first ûj(k) and secon moments using the recurrent procedure (Eqs.14 to 16) (Evlanov and Zhuk 1990) ) u k P k are the expected value and correlation matrix of conditional a poste Wj(u(k)), refined by measurements ( ), 1, s k s l  = .In this case the initial conditions for the proced ( 1) / ( ) ˆ( 1) / ( ).
Equation of calculation of conditional a posteriori PD Wj(u(k)) (Eq.10) at sequentially processing of the arriving measurements ( ), 1, l k l S  = comes down to calculation of its first ûj(k) and second ˆ( ) j P k moments using the recurrent procedure (Eqs.14 to 16) (Evlanov and Zhuk 1990) ) where: ( ), ( ) u k P k are the expected value and correlation matrix of conditional a posteriori PD Wj(u(k)), refined by measurements ( ), 1, s k s l  = .In this case the initial conditions for the procedure Eqs. .The advantage of this method is the possibility to organize processing with limited computing resources.It is especially convenient in case of nonsynchronous moments of input data arrival in comparison with algorithms of parallel processing (Evlanov and Zhuk 1990;Tovkach and Zhuk 2017c).
The algorithm of the filtration of discrete components does not change and is described by the Eqs.8 and 11.
In this case conditional PD P(uΔ(k) | aj(k), UΔ(k -1)) = N(H(u * j(k)), Dj(k)) is Gaussian, and correlation matrix Dj(k) is determined by Eq. 17 The quasi-optimal device that implements the algorithm Eqs. 8, 11 to 16 is also multichannel with number of channels M and generally keeps the structure of the optimal device.Their difference is that in quasi-optimal device only the first and second moments of conditional PD W * j(u(k)) and Wj(u(k)) are calculated.In this case the quasi-optimal algorithm allows keeping representation of a posteriori PD Wj(u(k)) in the form of the sum of the M Gaussian densities.
The block diagram of the j-th channel of such quasi-optimal filter is shown in Fig. 1.
where u ˆlj (k), P ˆlj (k) are the expected value and correlation matrix of conditional a posteriori PD W j (u(k)), refined by measurements Δ s (k), s = 1, l.In this case the initial conditions for the procedure Eqs. 14 to 16 at l = 0 has the form u ˆ0j (k) = u ˆ*j (k), P ˆ0j (k) = P ˆ*j (k), j = 1, M, and u ˆj(k) = u ˆLj (k), P ˆj(k) = P ˆLj (k), j = 1, M,.The advantage of this method is the possibility to organize processing with limited computing resources.It is especially convenient in case of nonsynchronous moments of input data arrival in comparison with algorithms of parallel processing (Evlanov and Zhuk 1990;Tovkach and Zhuk 2017c).
The algorithm of the filtration of discrete components does not change and is described by the Eqs.8 and 11.In this case conditional PD P(u Δ (k) | a j (k), U Δ (k -1)) = N(H(u * j (k)), D j (k)) is Gaussian, and correlation matrix D j (k) is determined by Eq. 17: The quasi-optimal device that implements the algorithm Eqs. 8, 11 to 16 is also multichannel with number of channels M and generally keeps the structure of the optimal device.Their difference is that in quasi-optimal device only the first and second moments of conditional PD W * j (u(k)) and W j (u(k)) are calculated.In this case the quasi-optimal algorithm allows keeping representation of a posteriori PD W j (u(k)) in the form of the sum of the M Gaussian densities.
The block diagram of the j-th channel of such quasi-optimal filter is shown in Fig. 1.
Figure 1.Block diagram of one channel quasi-optimal filter at consecutive calculations.

THE EFFECTIVENESS OF THE ALGORITHM ANALYSIS
The operability of the obtained quasi-optimal adaptive algorithm of the UAV movement parameters estimate on the basis of TDOA measurements Eqs. 8, 11 to 16 is checked by the Monte Carlo method.
The partial derivatives included in Eqs. 14 to 17 have the form estimates the distance between l-th sensor and UAV, estimates the distance between the reference sensor and the UAV.
The initial conditions (0), (0), 1,3 where: 1 ˆˆ ( ( the hovering and almost uniform motion of the UAV is close to unity probability of recognizing the maneuver is lower due to its short duration.

2
; 1 , , a a a cceleration and speed of change of acceleration of the UAV, 5 m/s; a2 = 0.01 m/s 2 ; a3 = 6 m/s 3 .where: for the first hypothesis j = 1 were created on the current the observations at the two neighboring steps.
recognition probability of movement of the first, second and ained by statistical modeling.The probability of recognizing f the UAV is close to unity by the algorithm obtained.The ower due to its short duration.
are RMS of random fluctuations of speed, acceleration and speed of change of acceleration of the UAV, respectively.When modeling relied: a1 = 0.05 m/s; a2 = 0.01 m/s 2 ; a3 = 6 m/s 3 .
The partial derivatives  where estimates the distance between l-th sensor and UAV, l = 1,S; estimates the distance between the reference sensor and the UAV.
The initial conditions u ˆj(0), P ˆj(0), j = 1,3 for the first hypothesis j = 1 were created on the current measurements, and for j = 2,3 according to the observations at the two neighboring steps.
Figure 3 shows the dependences of recognition probability of movement of the first, second and third types, respectively, curves 1, 2, 3, obtained by statistical modeling.The probability of recognizing the hovering and almost uniform motion of the UAV is close to unity by the algorithm obtained.The probability of recognizing the maneuver is lower due to its short duration.
Figure 4 shows dependences of expected value (curve 1) and RMS (curve 2) errors of the position estimation of the UAV along coordinates of X, Y, Z and also RMS (curve 3) errors of assessment calculated by the adaptive algorithm obtained experimentally.Figure 4  Algorithms of estimating the UAV movement parameters using two Kalman filters using models of almost uniform motion j = 2 and motion with maneuver j = 3 were also investigated.Estimation errors of the UAV position of the Kalman filter based on the model j = 2 (Fig. 5) contain systematic components, which in some sections of the trajectory are 2 times higher than the RMS error of the position estimation due to the presence of maneuvers.The developed adaptive algorithm allows to increase the accuracy of the UAV movement parameters estimate on hover and motion without maneuver sections in comparison with the Kalman filter based on the model j = 3 (Fig. 6) more than 2 to 3 times.

CONCLUSIONS
The optimal algorithm of adaptive filtration of UAV motion parameters based on TDOA measurements (Eqs.8 to 11) obtained on the basis of a mathematical apparatus of mixed Markov processes in discrete time allows to recurrently calculate the joint PD of a continuously-valued vector and the switching variable at each step.The optimal device that realizes the algorithm (Eq.8 to 11) has M channels.There is no growth of channels number due to the Markov property of the mixed process.
In the quasi-optimal algorithm of adaptive filtration of UAV movement parameters based on TDOA measurements (Eqs.8, 11 to 16) obtained by linearization of the equation of measurement (Eq.3), Gaussian approximation of conditional a posteriori distributions of a continuous-valued component is carried out.At the same time, it estimates the measurement error of the reference sensor, which allows to realize serial processing of measurements and to avoid the inversion of large-dimensional matrices.
Application of a trajectory filtration allows to reduce RMS of errors of definition of the UAV location in comparison with RMS of errors of location definition by TDOA method (the lower bound of Cramer-Rao) by 2 to 4 times.For the reviewed example the developed adaptive algorithm allows to increase the accuracy of estimating the UAV movement parameters on hover and motion without maneuver sections in comparison with the Kalman filter based on the model j = 3 more than 2 to 3 times and to avoid the occurrence of systematic error estimates.At the same time, the probability of recognizing these areas is close to unity.

G
are the matrixes describing various types ω tr (k) is the uncorrelated sequence of Gaussian vectors with a single correlation matrix.
process W(u(k), aj(k)) = P(u(k), aj(k) | UΔ(k)), where UΔ(k) = uΔ(1),…,uΔ(k) are the sequences of ents received to till k-th moment, included.Introducing also designation of the extrapolated PD j(k) | UΔ(k -1)) for the expanded process and following the technique of synthesis(Zhuk 1989), al algorithm of a filtration can be presented in the form of two the recurrent Eqs.6 are the sequences of measurements received to till k-th moment, included.Introducing also designation of the extrapolated PD P(u(k), aj(k) | UΔ(k -1)) for the expanded process and following the technique of synthesis(Zhuk 1989), the optimal algorithm of a filtration can be presented in the form of two the recurrent Eqs.
are the sequences of nts received to till k-th moment, included.Introducing also designation of the extrapolated PD ) | UΔ(k -1)) for the expanded process and following the technique of synthesis(Zhuk 1989), l algorithm of a filtration can be presented in the form of two the recurrent Eqs.
u k W u k a k du k Equation 6 describes evolution of the extrapolated PD W * (u(k), aj(k)) and is also an optimal algorithm of extrapolation of mixed Markov processes (u(k), aj(k)) for one step.With the help of Eq. 7 the clarifying of extrapolated PD based on the obtained measurement uΔ(k) is made and a posteriori PD The problem of synthesis of an optimal algorithm comes down to calculation of ap expanded process W(u(k), aj(k)) = P(u(k), aj(k) | UΔ(k)), where UΔ(k) = uΔ(1),…,uΔ(k) are t measurements received to till k-th moment, included.Introducing also designation of the e P(u(k), aj(k) | UΔ(k -1)) for the expanded process and following the technique of synthes the optimal algorithm of a filtration can be presented in the form of two the recurrent Eqs.

Figure 2 .
Figure 2. Configuration of the sensor network with nine sensors and the trajectory of UAV movement.

Figure 2 .
Figure 2. Configuration of the sensor network with nine sensors and the trajectory of UAV movement.
a a are RMS of random fluctuations of speed, acceleration and speed of change of acceleration of the UAV, respectively.When modeling relied: a1 = 0.05 m/s; a2 = 0.01 m/s 2 ; a3 = 6 m/s 3 . .14 to 17 have the form where F b j , G b j and j = 1,3 have the form are RMS of random fluctuations of speed, acceleration and speed of change of acceleration of the UAV, respectively.When modeling xx/xx 07/10 relied: a 1 = 0.05 m/s; a 2 = 0.01 m/s 2 ; a 3 = 6 m/s 3 .The partial derivatives дh l (u ˆ j l-1 hypothesis j = 1 were created on the current measurements, and for 2,3 j = according to the observations at the two neighboring steps.

Figure 3
Figure 3 shows the dependences of recognition probability of movement of the first, second and third types, respectively, curves 1, 2, 3, obtained by statistical modeling.The probability of recognizing the hovering and almost uniform motion of the UAV is close to unity by the algorithm obtained.The probability of recognizing the maneuver is lower due to its short duration.
Figure3shows the dependences of recognition probability of mo third types, respectively, curves 1, 2, 3, obtained by statistical modeling.
hypothesis j = 1 were created on the current measurements, and for 2,3 j = according to the observations at the two neighboring steps.

Figure 3
Figure 3 shows the dependences of recognition probability of movement of the first, second and third types, respectively, curves 1, 2, 3, obtained by statistical modeling.The probability of recognizing the hovering and almost uniform motion of the UAV is close to unity by the algorithm obtained.The probability of recognizing the maneuver is lower due to its short duration.
Figure 4 shows dependences of expected value (curve 1) and RMS (curve 2) errors of the position estimation of the UAV along coordinates of X, Y, Z and also RMS (curve 3) errors of assessment calculated by the adaptive algorithm obtained experimentally.Figure 4 also shows dependences of RMS error of measurement of the UAV position, which corresponds to the lower bound of Cramer-Rao (curve 4).Accumulation of measurements of a trajectory filtration allows to reduce RMS error of location definition of the UAV in comparison with the lower bound of Cramer-Rao (Tovkach and Zhuk 2017b) by 2 to 4 times.Algorithms of estimating the UAV movement parameters using two Kalman filters using models of almost uniform motion j = 2 and motion with maneuver j = 3 were also investigated.Estimation errors of the UAV position of the Kalman filter based on the model j = 2 (Fig.5) contain systematic components, which in some sections of the trajectory are 2 times higher than the RMS error of the position estimation due to the presence of maneuvers.The developed adaptive algorithm allows to increase the accuracy of the UAV movement parameters estimate on hover and motion without maneuver sections in comparison with the Kalman filter based on the model j = 3 (Fig.6) more than 2 to 3 times.

Figure 5 .Figure 6 .
Figure 5. RMS errors of the UAV position estimate using Kalman filter based on the model of j = 2.