A priori estimation based trap method for unmanned aerial vehicle trajectory deception

Abstract

The application discloses a trap type unmanned aerial vehicle track deception method based on prior estimation, relates to the technical field of unmanned aerial vehicles, and sets a deception trap in the possible coming direction of a black flight unmanned aerial vehicle, so that when the black flight unmanned aerial vehicle moves into the trap range, a deception signal controls the tracking loop of the unmanned aerial vehicle, and the unmanned aerial vehicle is lured to a specified area. The trap type track deception technology adopted by the application does not need to use radar and other detection equipment to obtain the motion parameters of a target in real time, thereby reducing the hardware cost. Moreover, the track deception algorithm provided by the application generates deception position parameters and deception speed parameters of a deception signal through a greedy strategy, does not need to estimate the control loop of the unmanned aerial vehicle, reduces the technical complexity, and is more in line with the actual situation.

Description

TECHNICAL FIELD

[0001] The present application relates to the technical field of unmanned aerial vehicles, in particular to a trap type unmanned aerial vehicle trajectory deception method based on priori estimation. BACKGROUND

[0002] With the rapid increase of the market size of unmanned aerial vehicles, incidents of unmanned aerial vehicles threatening public safety occur from time to time. At present, the main idea of countermeasures against black flight unmanned aerial vehicles is to make black flight unmanned aerial vehicles land or move to a designated area through electromagnetic interference. This idea can be divided into two categories. One is suppression interference, which cuts off the communication link of the unmanned aerial vehicle by transmitting a directional high-power radio frequency interference signal to the unmanned aerial vehicle, forcing the unmanned aerial vehicle to land or return. The other is deception interference. Currently, civil unmanned aerial vehicles mainly use combined navigation technology of satellite navigation and inertial navigation (GNSS / INS) for positioning. Since the satellite navigation signal is very weak when it reaches the ground, the deception interference technology uses a fake satellite navigation signal to deceive the unmanned aerial vehicle, so that it deviates from the preset trajectory and lands in a designated area.

[0003] Since the suppression interference technology cannot control the black flight unmanned aerial vehicle, and the high-power radio frequency interference signal will indiscriminately affect other civil facilities, the current research focus is mainly on the deception interference technology. The deception interference technology makes the positioning result of the unmanned aerial vehicle output an error value through GNSS deception interference technology, thereby causing the unmanned aerial vehicle to deviate from the trajectory. The deception interference technology can be divided into two categories. One category is to deceive only the GNSS positioning and speed measurement result of the unmanned aerial vehicle. This method assumes that the positioning and speed measurement result output by the GNSS receiver of the unmanned aerial vehicle is the state estimation of the unmanned aerial vehicle itself, and does not consider the data fusion of other sensors of the unmanned aerial vehicle. If this method is directly applied to a combined navigation unmanned aerial vehicle, it may cause the deception deception to fail or the actual deceived trajectory of the target to be unexpected. The other category of technology considers the combined navigation model of the unmanned aerial vehicle. When performing trajectory deception, the combined navigation filter gain of the unmanned aerial vehicle needs to be estimated, and then the generated deception signal is corrected.

[0004] However, the above-mentioned deception interference technology needs to obtain the relatively accurate position, speed and other parameters of the unmanned aerial vehicle in real time through radar and other detection devices, and needs to estimate the control loop of the unmanned aerial vehicle, resulting in high technical complexity and high hardware cost.

[0005] Therefore, how to provide a simple, economical and efficient unmanned aerial vehicle trajectory deception method is a problem that those skilled in the art need to solve. SUMMARY

[0006] Therefore, the application provides a trap type unmanned aerial vehicle trajectory deception method based on prior estimation, which does not need to use radar and other detection equipment to obtain the motion parameters of the target in real time, reduces the hardware cost, and the trajectory deception algorithm generates the deception position and speed of the deception signal through a greedy strategy, without the need to estimate the control loop of the unmanned aerial vehicle, thereby reducing the technical complexity and being more in line with the actual situation.

[0007] In order to achieve the above-mentioned purpose, the application adopts the following technical scheme: a trap type unmanned aerial vehicle trajectory deception method based on prior estimation, and the specific steps include:

[0008] Step 1: determining the spatial range of the deception trap, predicting the initial position and speed of the unmanned aerial vehicle;

[0009] Step 2: estimating the combined navigation Kalman model of the unmanned aerial vehicle to obtain the estimated value K of the Kalman filter steady-state gain s ;

[0010] Step 3: estimating the position and speed of the unmanned aerial vehicle at the k time, and estimating the prior estimation value of the combined navigation output of the unmanned aerial vehicle;

[0011] Step 4: obtaining the position deception parameter and speed deception parameter of the deception signal by using a greedy strategy, and generating a deception signal to invade the unmanned aerial vehicle;

[0012] Step 5: updating the estimated value of the combined navigation output of the unmanned aerial vehicle, and judging whether the unmanned aerial vehicle has reached the deception target area; if the unmanned aerial vehicle has reached the deception target area, the trajectory deception process is ended, otherwise the process of steps 3-5 is repeated.

[0013] Preferably, the spatial range of the deception trap is the main lobe shape of the deception jammer transmitting antenna, and the size of the deception trap is determined by the transmission power of the deception signal; the direction with the maximum gain of the deception jammer transmitting antenna is aligned to the black flight unmanned aerial vehicle, and the transmission power of the deception signal is set to determine the spatial range of the deception trap.

[0014] Since the position of the deception jammer is known, after the direction with the maximum gain of the deception jammer transmitting antenna is aligned to the black flight unmanned aerial vehicle and the transmission power of the deception signal is set, the spatial range of the deception trap can be determined, and then the position and speed of the unmanned aerial vehicle when entering the boundary of the deception trap can be estimated according to the relative relationship of the positions. The trap type deception can effectively counteract the black flight unmanned aerial vehicle and protect the safety of the key infrastructure.

[0015] Preferably, in step 2, the combined navigation Kalman model of the unmanned aerial vehicle is estimated to obtain the estimated value K of the Kalman filter steady-state gain sThe specific content includes: according to the Kalman filtering process, after iteration, the Kalman gain matrix K will enter the steady state K ∞ ; wherein K ∞ is determined by the system state transition matrix A, the noise matrix Q, the observation noise matrix R and the observation matrix H, and the estimated value K s of the Kalman filtering steady-state gain is obtained by an iterative method according to the determined system state transition matrix A, the noise matrix Q, the observation noise matrix R and the observation matrix H.

[0016] Since the civil unmanned aerial vehicle usually adopts GNSS / INS integrated navigation for navigation and positioning, the GNSS / INS integrated navigation will perform data fusion on the position and speed information output by the GNSS and the position and speed calculated by the INS through Kalman filtering. When the GNSS spoofing signal is used to deceive the trajectory of the unmanned aerial vehicle, the false position and speed output by the integrated navigation system of the unmanned aerial vehicle is not the false position and speed output by the GNSS receiver of the unmanned aerial vehicle caused by the GNSS spoofing signal, and the fusion of the GNSS and the INS by the Kalman filtering process also needs to be considered, so it is necessary to estimate the Kalman filtering gain of the GNSS / INS integrated navigation of the unmanned aerial vehicle.

[0017] Preferably, in step 3, the position speed information of the unmanned aerial vehicle at time k is obtained according to the position, speed and acceleration of the unmanned aerial vehicle at time k-1.

[0018]

[0019]

[0020] In the formula, are respectively the position, speed and acceleration estimated values of the unmanned aerial vehicle at time k-1 by the spoofing interference source;

[0021] The Kalman filtering process of the unmanned aerial vehicle is simulated to obtain the position and speed posteriori estimated values of the integrated navigation output by the unmanned aerial vehicle at time k-1, and the position speed a priori estimated values of the integrated navigation output by the unmanned aerial vehicle at time k are obtained according to the position and speed posteriori estimated values of the integrated navigation output by the unmanned aerial vehicle at time k-1.

[0022]

[0023]

[0024] In the formula, are respectively the position and speed posteriori estimated values of the integrated navigation output by the unmanned aerial vehicle at time k-1.

[0025] Assuming that the UAV does uniform acceleration motion, the position and speed of the UAV at k time can be estimated according to the position, speed and acceleration of the UAV at k-1 time, and the prior estimation value of the combined navigation output at k time can be calculated according to the posterior estimation value of the combined navigation output at k-1 time.

[0026] Preferably, in step 4, the deception signal makes the position and speed output by the combined navigation system of the UAV deviate through the greedy strategy, and the direction of the UAV will point to the deception target area through the action of the acceleration component.

[0027] According to the position of the deception target area, the deception signal makes the position and speed output by the combined navigation system of the UAV deviate through the greedy strategy, and the deviation will make the UAV generate an acceleration component towards the deception target area, and after a long time of action of the acceleration component, the UAV will be lured to the specified area.

[0028] Preferably, the deception signal makes the position and speed output by the combined navigation system of the UAV deviate through the greedy strategy, and the direction of the UAV will point to the deception target area through the action of the acceleration component. speed information, the position and speed output by the combined navigation at k time are estimated according to the position speed of the UAV at k time end,s , and the prior estimation value of the combined navigation output at k time and the speed deception parameter

[0029]

[0030]

[0031] In the formula, eph is the horizontal position error tolerance of the combined navigation, evh is the horizontal speed error tolerance of the combined navigation,

[0032] Δd=k1.d1+k2.d2

[0033]

[0034]

[0035] k1 and k2 are adjustment coefficients, d1 is a unit vector pointing to the deception target area by estimating the position of the UAV, d2 is a reverse unit vector for estimating the speed of the UAV, and Δd is the sum of the two vectors.

[0036] The false position and speed generated by the deception signal generated by the greedy strategy acts on the UAV, and the control loop of the UAV, in particular the PD controller parameters, does not need to be estimated, which reduces the technical complexity and is more in line with the actual situation.

[0037] Preferably, in step 5, the updating of the estimated value of the combined navigation output of the UAV specifically comprises: simulating the Kalman filtering process of the UAV to obtain the position and speed posteriori estimation value of the combined navigation output of the UAV at time k:

[0038]

[0039] Wherein, are the position and speed posteriori estimation value of the UAV at time k, respectively, are the position and speed priori estimation value of the combined navigation output of the UAV at time k, respectively, represents the deception position, represents the deception speed, K s is the estimated value of the Kalman filtering steady-state gain.

[0040] The position and speed of the UAV are predicted by priori estimation, and the estimation value of the combined navigation output of the UAV is posteriori corrected according to the false positioning and speed information generated by the deception signal acting on the UAV, without the need to use radar and other detection equipment, thereby reducing the technical complexity and hardware cost.

[0041] According to the above technical solution, compared with the prior art, the present application provides a trap type UAV trajectory deception method based on priori estimation, and the specific steps include: determining the spatial range of the deception trap, predicting the initial position and speed of the UAV; estimating the combined navigation Kalman model of the UAV to obtain the estimated value K s of the Kalman filtering steady-state gain; estimating the position and speed of the UAV at time k, and estimating the priori estimation value of the combined navigation output of the UAV; obtaining the position deception parameter and the speed deception parameter of the deception signal by using the greedy strategy, and generating the deception signal to invade the UAV; updating the estimation value of the combined navigation output of the UAV, and judging whether the UAV has reached the deception target area; if the UAV has reached the deception target area, the trajectory deception process is ended, otherwise the above process is repeated.

[0042] The beneficial effects of this invention are as follows: This invention can deceive unauthorized drones flying in a certain direction within a certain range, thereby protecting critical infrastructure. The method involves setting a deception trap in the possible direction of the unauthorized drone. When the drone moves into the trap's range, a deception signal controls the drone's tracking loop, luring the drone to a designated area. The trap-based trajectory deception technology used in this method does not require real-time acquisition of the target's motion parameters using radar or other detection equipment, reducing hardware costs. Furthermore, the trajectory deception algorithm proposed in this invention generates the deception position and velocity parameters of the deception signal through a greedy strategy, eliminating the need to estimate the drone's control loop, reducing technical complexity and better reflecting real-world conditions. Attached Figure Description

[0043] To more clearly illustrate the technical solutions in the embodiments of the present invention or the prior art, the drawings used in the description of the embodiments or the prior art will be briefly introduced below. Obviously, the drawings described below are only embodiments of the present invention. For those skilled in the art, other drawings can be obtained based on the provided drawings without creative effort.

[0044] Figure 1 A flowchart of a trap-type UAV trajectory deception method based on prior estimation provided by the present invention;

[0045] Figure 2 This is a schematic diagram illustrating the use of a greedy strategy to obtain the deception position and deception speed, as provided by the present invention.

[0046] Among them, 1-the location p of the deception target area end,s 2-UAV position estimation at time k 3 - Combined navigation position error tolerance; 4 - Combined navigation speed error tolerance. Detailed Implementation

[0047] The technical solutions of the embodiments of the present invention will be clearly and completely described below with reference to the accompanying drawings. Obviously, the described embodiments are only some embodiments of the present invention, and not all embodiments. Based on the embodiments of the present invention, all other embodiments obtained by those skilled in the art without creative effort are within the scope of protection of the present invention.

[0048] This invention discloses a trap-based drone trajectory deception method based on prior estimation, such as... Figure 1 As shown, the specific steps include:

[0049] Step 1: Determine the spatial range of the deception trap and predict the initial position and velocity of the drone. Specifically, this is implemented as follows:

[0050] The spatial range of the deception trap can be approximately shaped as the main lobe of the transmitting antenna of the deception jammer, and the size is determined by the transmitting power of the deception signal. When a satellite navigation receiver is located within the spatial range of the deception trap, the deception signal will control the tracking loop of the receiver and make it output false position and speed information. When a black flying UAV is found, the direction with the maximum gain of the transmitting antenna of the deception jammer is aligned to the black flying UAV, and it is assumed that the direction of arrival of the UAV is the direction with the maximum gain of the transmitting antenna of the deception jammer. Since the resultant force of the acceleration of the UAV and the air resistance is zero when the UAV is flying steadily, it can be assumed that the UAV is moving at a constant speed in a straight line at this time. The speed of the UAV is v. Since the position of the deception jammer is known, after the direction with the maximum gain of the transmitting antenna of the deception jammer is aligned to the black flying UAV and the transmitting power of the deception signal is set, the spatial range of the deception trap can be determined, and then the initial position and speed of the UAV when it enters the boundary of the deception trap can be estimated according to the relative relationship of the positions. In the northeast celestial coordinate system established with the position of the deception jammer as the origin, it is assumed that the angle between the direction with the maximum gain of the transmitting antenna of the deception jammer and the north direction is θ, the farthest end point of the deception trap corresponds to the direction with the maximum gain of the transmitting antenna of the deception jammer, and the distance between the farthest end point of the deception trap and the deception jammer is L. Then the position speed of the UAV in the XOY two-dimensional plane of the coordinate system are respectively:

[0051]

[0052] In one specific embodiment of the present application, the typical value of the moving speed of a civilian multi-rotor UAV is 20 m / s. According to the above description, in the northeast celestial coordinate system established with the position of the deception jammer as the origin, the position speed of the UAV in the XOY two-dimensional plane of the coordinate system are respectively:

[0053]

[0054] Step 2: estimate the combined navigation Kalman model of the UAV to obtain the estimated value K s of the steady-state gain of the Kalman filter, which is specifically implemented as:

[0055] Since the civil unmanned aerial vehicle usually adopts GNSS / INS integrated navigation for navigation and positioning, the GNSS / INS integrated navigation fuses the position and speed information output by the GNSS and the position and speed calculated by the INS through Kalman filtering. When the GNSS spoofing signal is used to deceive the trajectory of the unmanned aerial vehicle, the error position and speed output by the integrated navigation system of the unmanned aerial vehicle are not the error position and speed output by the GNSS receiver of the unmanned aerial vehicle caused by the GNSS spoofing signal, and the fusion of the GNSS and the INS in the Kalman filtering process also needs to be considered, so it is necessary to estimate the Kalman filtering gain of the GNSS / INS integrated navigation of the unmanned aerial vehicle. According to the Kalman filtering process, after a plurality of iterations, the Kalman gain matrix K will enter a steady state K ∞ 。K ∞ which is determined by the system state transition matrix A, the noise matrix Q, the observation noise matrix R and the observation matrix H. After A, Q, R and H are determined, the estimated value K s of the Kalman filtering gain is obtained through the iteration method.

[0056] Step 3: estimating the position and speed of the unmanned aerial vehicle at the k time and estimating the priori estimation value output by the integrated navigation of the unmanned aerial vehicle, which is specifically implemented as:

[0057] According to the typical Kalman filtering model of the integrated navigation, the position and speed output by the integrated navigation of the unmanned aerial vehicle are estimated. In an embodiment of the present application, when the spoofing signal is generated and acts on the unmanned aerial vehicle, the unmanned aerial vehicle will find that its positioning deviates from the preset route, at which time the unmanned aerial vehicle will generate an acceleration to adjust its position and speed. Assuming that the unmanned aerial vehicle moves at a uniform acceleration, the position of the unmanned aerial vehicle at the k-1 time can be expressed as The speed is The acceleration is From which the position of the unmanned aerial vehicle at the k time is calculated as The speed is

[0058]

[0059] In the formula, is the estimation value of the position, speed and acceleration of the unmanned aerial vehicle at the k-1 time by the spoofing interference source.

[0060] According to the posteriori estimation value output by the integrated navigation of the unmanned aerial vehicle at the k-1 time, the position The speed is The priori estimation value is:

[0061]

[0062] In the formula, The position and speed posterior estimation value of the unmanned aerial vehicle combination navigation output at k-1 moment.

[0063] Step 4: The position deception parameter and the speed deception parameter of the deception signal are obtained by using the greedy strategy, and the deception signal invades the target unmanned aerial vehicle, and the specific implementation is as follows:

[0064] The position deception parameter and the speed deception parameter of the deception signal are obtained by using the greedy strategy, and the deception signal invades the target unmanned aerial vehicle, and the specific implementation is as follows: end,s According to the protected target area, the safe position away from the protected target area is determined as the position p end,s of the deception target area, and according to the position of the deception target area, the position and speed output by the unmanned aerial vehicle combination navigation system are deviated by using the greedy strategy, and the deviation will make the unmanned aerial vehicle generate an acceleration component towards the deception target area, and after a long time of the acceleration component, the speed of the unmanned aerial vehicle will finally point to the deception target area, as shown in Figure 2 According to the position speed estimation value of the unmanned aerial vehicle at k moment, the prior estimation value output by the unmanned aerial vehicle combination navigation and the position p end,s of the deception target area, the position deception parameter and the speed deception parameter are obtained by using the greedy strategy.

[0065]

[0066] Wherein, eph is the horizontal position error tolerance of the combination navigation, and evh is the horizontal speed error tolerance of the combination navigation. In the formula:

[0067]

[0068] Wherein, k1 and k2 are adjustment coefficients, d1 is a unit vector pointing to the deception target area according to the position estimation of the unmanned aerial vehicle, d2 is a unit vector opposite to the speed estimation of the unmanned aerial vehicle, and Δd is the sum of the two vectors.

[0069] Step 5: The estimation value of the unmanned aerial vehicle combination navigation output is updated, and it is judged whether the unmanned aerial vehicle reaches the deception target area or not; if the unmanned aerial vehicle has reached the deception target area, the trajectory deception process is ended, otherwise the process of steps 3-5 is repeated. The specific implementation process is as follows:

[0070] According to the error positioning and speed information generated by the deception signal acting on the unmanned aerial vehicle, the estimation value of the unmanned aerial vehicle combination navigation output is posteriorly corrected: the Kalman filtering process of the unmanned aerial vehicle is simulated to obtain the position and speed posterior estimation value of the unmanned aerial vehicle combination navigation output at k moment:

[0071]

[0072] wherein, respectively are the position and velocity posterior estimates of the UAV at time k from the combined navigation output of the deceptive jammer, respectively are the position and velocity prior estimates of the UAV at time k from the combined navigation output, denotes the deceptive position, denotes the deceptive velocity, K s is the estimate of the steady-state gain of the Kalman filter.

[0073] For

[0074]

[0075] At the same time, it is judged whether the UAV has arrived at the deceptive target area.If the UAV has arrived at the target deceptive area, the trajectory deception process is ended, otherwise the process of steps 3-5 is repeated.

[0076] The present application provides a trap type UAV trajectory deception method based on prior estimation, by setting a deception trap in the possible direction of the black UAV, when the black UAV moves into the trap range, the deception signal will control the tracking loop of the UAV, so that the UAV is lured to the designated area, the black UAV in a certain range of direction can be trajectory lured, so as to protect the key infrastructure.

[0077] The embodiments in the specification are described in a progressive manner, and each embodiment focuses on the difference from other embodiments, and the same or similar parts between the embodiments can be referred to each other. For the device disclosed in the embodiments, since it corresponds to the method disclosed in the embodiments, the description is relatively simple, and the related parts can be referred to the method part.

[0078] The above description of the disclosed embodiments enables a person skilled in the art to implement or use the present application. Various modifications to the embodiments will be apparent to those skilled in the art, and the general principles defined herein can be implemented in other embodiments without departing from the spirit or scope of the present application. Therefore, the present application will not be limited to the embodiments shown herein, but will conform to the widest scope consistent with the principles and novel features disclosed herein.

Claims

1. A priori estimation based trap drone trajectory deception method, characterized in that, The specific steps include: Step 1: determining the spatial range of the deception trap, predicting the initial position and speed of the UAV; Step 2: Estimate the combined navigation Kalman model of the UAV to obtain an estimated value of the Kalman filter steady-state gain ; Step 3: Estimate the position and velocity of the UAV at the moment, and estimate the prior estimate value of the UAV combined navigation output; Step 3: Estimate the position and velocity of the UAV at the moment, and estimate the prior estimate value of the UAV combined navigation output; Step 4: obtaining the position deception parameter and the speed deception parameter of the deception signal by using the greedy strategy, and generating the deception signal to invade the UAV; Step 5: updating the estimated value of the output of the UAV integrated navigation, and judging whether the UAV has reached the deception target area; if the UAV has reached the deception target area, the trajectory deception process is ended, otherwise the process of steps 3-5 is repeated; In step 4, the position and speed output by the UAV integrated navigation system are deviated by the deception signal through the greedy strategy, and the direction of the UAV is directed to the deception target area through the action of the acceleration component; The position and speed output by the unmanned aerial vehicle integrated navigation system are deviated by the deception signal through the greedy strategy, specifically including: according to the position and speed information of the unmanned aerial vehicle at the moment, prior estimation of the position and speed output by the unmanned aerial vehicle integrated navigation system at the moment, and the deception target area, adopting the greedy strategy to obtain the position deception parameter and the speed deception parameter. ​​​​​​​​​ wherein is a horizontal position error tolerance for the integrated navigation, is a horizontal velocity error tolerance for the integrated navigation, 、 is a tuning coefficient, denotes a unit vector pointing towards the deception target area for the UAV position estimate, denotes a unit vector pointing in the opposite direction of the UAV velocity estimate, is the sum of the two vectors.

2. The priori estimation based trap method for drone trajectory deception according to claim 1, wherein, The spatial range of the deception trap is the main lobe shape of the deception jamming source transmitting antenna, and the size of the deception trap is determined by the transmission power of the deception signal; the direction with the maximum gain of the deception jamming source transmitting antenna is aligned to the black flight UAV, and the transmission power of the deception signal is set to determine the spatial range of the deception trap.

3. The method of claim 1, wherein, In step 2, the combined navigation Kalman model of the unmanned aerial vehicle is estimated to obtain an estimated value of Kalman filter steady-state gain The specific content includes: according to the Kalman filter process, after iteration, the Kalman gain matrix will enter a steady state ; wherein determined by the system state transition matrix , noise matrix , observation noise matrix , observation matrix , according to the determined system state transition matrix , noise matrix , observation noise matrix , observation matrix , the estimated value of the Kalman filter steady-state gain is obtained by iteration method .

4. The method of claim 1, wherein, In step 3, according to the position, speed, acceleration of the UAV at the moment, the position of the UAV at the moment is obtained information:​​​ In the formula, , , are respectively The moment of deception interferes with the position, speed, and acceleration estimates of the UAV. simulate the Kalman filtering process of the UAV to obtain position and speed posterior estimation values of the UAV output by the integrated navigation at time t, and obtain position and speed prior estimation values of the UAV output by the integrated navigation at time t according to the position and speed posterior estimation values of the UAV output by the integrated navigation at time t. time t.​ In the formula, , are respectively The position and speed posterior estimation values of the UAV combined navigation output at the moment.

5. The method of claim 1, wherein, In step 5, the update of the estimated value of the UAV integrated navigation output specifically includes: simulating the Kalman filtering process of the UAV to obtain the UAV's... Posterior estimates of position and velocity output by the combined navigation system at each moment: wherein , are the position and velocity posterior estimates of the UAV at time tk output by the integrated navigation system, , are the position and velocity prior estimates of the UAV at time tk output by the integrated navigation system, denotes the spoofed position, denotes the spoofed velocity, is the estimate of the Kalman filter steady-state gain.​​

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