Air route safety guarantee system for low-altitude economic unmanned aerial vehicle

Abstract

An air route safety guarantee system for a low-altitude economic unmanned aerial vehicle comprises a signal receiving module, a data processing module, a signal transmitting module and a loop correction module, the signal receiving module receives a real satellite navigation signal, and the real satellite navigation signal is output to the data processing module after being subjected to down-conversion; the data processing module monitors satellite navigation interference in the down-conversion signal in real time and generates a satellite navigation countering signal to control the target unmanned aerial vehicle, and the signal transmitting module performs digital-to-analog conversion and up-conversion processing on the satellite navigation countering signal and transmits the satellite navigation countering signal to the target unmanned aerial vehicle. And the loop correction module corrects the parameters of the satellite navigation countering signal through signal sorting, so that the satellite navigation countering signal keeps synchronous with the real signal. Through a dual protection mechanism, electromagnetic interference can be eliminated in real time in a key area to guarantee navigation safety, meanwhile, an accurate navigation counter signal is generated after an illegal unmanned aerial vehicle is detected, the illegal unmanned aerial vehicle is trapped to a designated place, and the air route safety is remarkably improved.

Description

Technical Field

[0001] The present invention relates to a technology in the field of UAV detection, specifically a low-altitude economic UAV route safety guarantee system. Background Art

[0002] In recent years, China has vigorously developed the low-altitude economy. As an important part, UAVs play a significant role, but their low-altitude safety problems are becoming increasingly prominent. Existing countermeasures for UAVs mainly fall into three categories: physical damage, interference blocking, and deception control. Physical damage uses precise strikes such as missiles / lasers, which are costly and prone to causing secondary harm, making them difficult to apply in the civilian field. Interference blocking forces UAVs to land through electromagnetic suppression, and deception control forges navigation signals to guide the flight path. However, both rely on high-power signals, which are likely to interfere with normal surrounding equipment and threaten the safety of low-altitude air routes. Existing countermeasure devices generally have problems such as high power consumption and short battery life, and it is difficult to meet the precision and low-interference requirements in large-scale low-altitude economic scenarios. Summary of the Invention

[0003] Aiming at the deficiency that the existing technology only includes countermeasure functions or monitoring functions, the present invention proposes a low-altitude economic UAV route safety guarantee system. Through a dual protection mechanism, it can clear electromagnetic interference in real time in key areas to ensure navigation safety. At the same time, after detecting illegal UAVs, it generates precise navigation countermeasure signals to lure them to a designated location, significantly improving the safety of air routes.

[0004] The present invention is realized through the following technical solutions:

[0005] The present invention relates to a low-altitude economic UAV route safety guarantee system, including: a signal receiving module, a data processing module, a signal transmitting module, and a loop correction module. Among them: the signal receiving module receives real satellite navigation signals, and after down-conversion, outputs them to the data processing module. The data processing module monitors satellite navigation interference in the down-converted signals in real time and generates satellite navigation countermeasure signals to control target UAVs. The signal transmitting module performs digital-to-analog conversion and up-conversion processing on the satellite navigation countermeasure signals and sends them to the target UAVs. The loop correction module corrects the parameters of the satellite navigation countermeasure signals through signal sorting to keep them synchronized with the real signals.

[0006] The described data processing module includes: an external interference signal detection unit and a navigation interference countermeasure signal generation unit, where: the external interference signal detection unit performs preheating and acquisition on the satellite signals received by the signal reception module in groups according to the orbital plane, and realizes the rapid detection of interference signals based on the relevant peak morphology and AGC value in the acquisition stage; the navigation interference countermeasure signal generation unit predicts key parameters such as the transmission time and Doppler frequency of the countermeasure signal according to the received real signal and information such as the position of the target UAV. After modifying the data code of the real signal, it modulates and generates the baseband signal of the countermeasure signal.

[0007] The described loop correction module includes: a loop correction unit and a signal selection unit, where: the loop correction unit controls the signal selection unit to select a signal loop for signal self-circulation measurement to obtain the internal delay τ D and feedback it to the data processing module; the signal selection unit processes the countermeasure signal generation loop or the signal correction loop according to the instruction of the loop correction unit.

[0008] The present invention relates to an interference detection method for low-altitude economic UAV route safety based on the above system, including:

[0009] Step 1: Group the satellites according to their orbital planes. GPS satellites are divided into 6 groups according to orbital planes ABCDEF, with the number of satellites being 4, 6, 5, 6, 6, and 5 respectively. BDS satellites are divided into 3 groups according to GEO, IGSO, and MEO orbital planes, with the number of satellites being 7, 10, and 27 respectively;

[0010] Step 2: Perform preheating and acquisition on all satellites. If the detection system is a single GPS or BDS system, all satellites are cyclically detected 3 times. If the detection system is a GPS / BDS joint system, GPS satellites are cyclically detected 2 times in the order of GPS, BDS, GPS, and BDS satellites are cyclically detected 1 time.

[0011] Step 3: After preheating is completed, establish a satellite acquisition status list according to the preheating results. The GPS system is established in the order of orbital planes A - F, and BDS is established in the order of GEO, IGSO, and MEO. For the satellites captured during the preheating process, mark their satellite acquisition identification as 1 and the counter as 5. For the satellites not captured, mark the satellite acquisition identification as 0 and the counter as 0.

[0012] Step 4: Continuously perform acquisition. Each round of acquisition is carried out again according to a certain number of captured satellites and a certain number of uncaptured satellites, and the acquisition results are divided into two cases, and the value of the counter is updated respectively.

[0013] Step 5: Perform double-peak detection on the captured satellites. If there are double peaks in the captured satellites, it is considered that there is interference. At the same time, detect the AGC value. If it is abnormal, it is considered that there is interference.

[0014] The present invention relates to a method for generating a navigation countermeasure signal for the route safety of low-altitude economic unmanned aerial vehicles based on the above-mentioned system, including:

[0015] Step 1: The signal receiving module receives the captured data of the RF front end and performs tracking demodulation to obtain the signal parameters and navigation message of the currently visible satellites.

[0016] Step 2: The navigation interference countermeasure signal generation unit constructs a countermeasure signal that maintains a high degree of time-frequency synchronization with the real signal according to the signal transmission time, Doppler frequency of the currently visible satellites, and the estimated position and speed information of the target unmanned aerial vehicle.

[0017] Step 3: Use the precise control of unmanned aerial vehicle electromagnetic interference countermeasure technology to determine the power level required for the generated countermeasure signal.

[0018] Step 4: The signal selection unit selects the loop correction unit to correct the parameters of the generated countermeasure signal to make it strictly synchronized with the real signal. Then, the corrected countermeasure signal is transmitted to the signal transmitting module and transmitted to the target unmanned aerial vehicle after upconversion. Technical effects

[0019] The satellite navigation real-time interference detection method based on the satellite orbital plane adopted by the present invention preheats and captures the satellites, performs grouped cyclic detection and establishes a satellite capture status list, and obtains the available satellite status through the maintenance of this list; the precise control of unmanned aerial vehicle electromagnetic interference countermeasure technology adopted forms an interference signal trap centered on the target unmanned aerial vehicle through the precise control of the frequency, phase, and power of the countermeasure signal. Only the navigation link of the unmanned aerial vehicle within the trap will be actually damaged, while the unmanned aerial vehicle outside the trap will not be affected even if it receives the interference signal. Compared with the prior art, the present invention can monitor the electromagnetic environment around the route of the unmanned aerial vehicle, quickly eliminate potential navigation interference, ensure the route navigation safety during the normal operation of the unmanned aerial vehicle, and can also countermeasure the target unmanned aerial vehicle invading the route of the unmanned aerial vehicle to ensure the route safety of the unmanned aerial vehicle. At the same time, the device is small in size and low in cost, which is convenient for large-scale application; it avoids complex satellite state operations, greatly improves the navigation interference detection efficiency and detection accuracy, and reduces the hardware cost; it ensures that other navigation terminals are not affected under the condition of successfully countering the target unmanned aerial vehicle; it supports satellite frequency points in the GPS and Beidou open signal formats. Description of the drawings

[0020] Figure 1 It is a schematic diagram of the system of the present invention;

[0021] Figure 2 It is a flowchart for detecting navigation signal interference;

[0022] Figure 3 It is a result diagram of an embodiment for detecting navigation signal interference

[0023] Figure 4 It is a flowchart for generating countermeasure signals against a target UAV;

[0024] Figure 5 It is a schematic diagram of the process for generating countermeasure signals;

[0025] Figure 6 It is a schematic diagram of a test connection for verifying the countermeasure signal generation function of the device;

[0026] Figure 7 It is a physical diagram of a test device for verifying the countermeasure signal generation function of the device;

[0027] Figure 8 It is a test result diagram for verifying the countermeasure signal generation function of the device. Specific implementation manners

[0028] As Figure 1 shown, this embodiment relates to a low-altitude economic UAV route safety guarantee system, including: a signal receiving module, a data processing module, a signal transmitting module, and a loop correction module.

[0029] As Figure 2 shown, this embodiment relates to an interference detection method for low-altitude economic UAV route safety based on the above system, including:

[0030] Step 1: Group the satellites according to their orbital planes. GPS satellites are divided into 6 groups according to orbital planes ABCDEF, with the number of satellites being 4, 6, 5, 6, 6, and 5 respectively. BDS satellites are divided into 3 groups according to GEO, IGSO, and MEO orbital planes, with the number of satellites being 7, 10, and 27 respectively. The specific grouping is shown in Table 1 and Table 2 below.

[0031] Table 1 GPS satellite orbital plane division Orbital plane Satellite number A G07, G24, G30, G31 B G12, G14, G16, G25, G26, G28 C G08, G17, G19, G27, G29 D G01, G02, G06, G11, G18, G21 E G03, G05, G10, G20, G22, G23 F G04, G09, G13, G15, G32

[0032] Table 2 BDS satellite orbital plane division

[0033] Step 2: Warm up and capture all the satellites. If the detection system is a single GPS or BDS system, cycle through all the satellites 3 times. If the detection system is a GPS / BDS joint system, cycle through the GPS satellites 2 times in the order of GPS, BDS, GPS, and detect the BDS satellites 1 time.

[0034] The so-called preheating capture is specifically as follows: Capture all currently visible satellites and record the information of the captured satellites. Since the number of visible satellites at a location is basically constant within a certain period of time, if the number of satellites suddenly increases at a certain moment, there may be artificial interference signals and further detection is required.

[0035] Step 3: After the preheating is completed, establish a satellite capture status list according to the preheating results. The GPS system is established in the order of A - F orbital planes in sequence, and the BDS is established in the order of GEO, IGSO, MEO in sequence. For the satellites captured during the preheating process, mark their satellite capture identifier as 1 and the counter as 5, and for the satellites not captured, mark the capture identifier as 0 and the counter as 0.

[0036] Step 4: Continuous capture, that is, update the counter according to the capture results of each round in different situations, specifically including: When satellites were captured in the previous round and not captured this time: Decrease the counter by 1, and if the counter is 0, set the capture identifier to 0; When satellites were captured in the previous round and captured this time: Set the counter to 5; When satellites were not captured in the previous round and not captured this time: The capture identifier and the counter remain unchanged; When satellites were not captured in the previous round and captured this time: Set the capture identifier to 1 and the counter to 5.

[0037] Step 5: While performing relevant peak detection on the captured satellites, detect the AGC value. If it is abnormal, it is considered that there is interference.

[0038] Regarding the relevant peak detection, as long as there are double peaks among the captured satellites, it is considered that there is interference.

[0039] The AGC is the automatic gain control, which is used to detect whether the signal reception power is abnormal, and the value can be read out by the GNSS receiver.

[0040] Through specific experiments, an interference signal is generated by a satellite signal spoofing interference signal generator for testing. According to the experimental results, as Figure 3 shown, this device can detect and record information such as the interference intensity, interference time, and the satellite numbers affected by the interference signal (red represents the existence of an interference signal, and green represents a normal signal).

[0041] As Figure 4 shown, this embodiment relates to a method for generating a navigation countermeasure signal for the low-altitude economy UAV route safety based on the above system, including:

[0042] Step 1: The navigation countermeasure signal generation unit is used to capture, track and demodulate the currently visible satellite signal to obtain the navigation message bit stream in subframe units, and then the transmission time, Doppler frequency and satellite status of the currently visible satellite signal are obtained through position, velocity and time (PVT) calculation.

[0043] Step 2: The navigation jamming countermeasure signal generation unit constructs a countermeasure signal that is highly synchronized with the real signal in terms of time and frequency based on the signal transmission time, Doppler frequency of the currently visible satellite and the estimated position and speed information of the target UAV, including:

[0044] 2.1 As Figure 5 As shown in the figure, the emission time and Doppler frequency parameters corresponding to the countermeasure signal are calculated according to the position of the countermeasure device and the position of the target drone, specifically: emission time in: is the launch time of the real visible satellite, is the pseudorange between the ith visible satellite and the device, is the pseudo-range between the ith visible satellite and the trapping target point, δt a is the local clock error of the device, δt b is the local clock error at the target point, L is the distance between the device and the target UAV, c is the speed of light, τ D is the inherent delay of the device, which is compensated by the loop correction module; Doppler frequency parameter in: is the visible satellite velocity, The speed required to capture the target point, is the unit vector between the visible satellite and the trapping target point, is the satellite clock drift, and λ is the wavelength of the satellite navigation signal.

[0045] 2.2 Update the second-of-week information of the navigation message according to the launch time obtained in step 2.1, then generate a new check code according to the navigation message encoding rules and the updated information and re-encode the message of the counter signal.

[0046] 2.3 Calculate the spread spectrum code phase, data code phase and carrier phase of each sampling point in turn, and calculate the value of each sampling point according to the signal structure to complete the navigation message and carrier modulation to generate a digital baseband signal.

[0047] Step 3: Use precision control of drone electromagnetic interference countermeasure technology to determine the power level required for the generated countermeasure signal, including:

[0048] 3.1 Use the step-by-step bias method to achieve covert capture of the target UAV. Determine the phase shift speed δτ of the countermeasure signal code and the initial code phase difference between the countermeasure signal and the real signal according to the position and speed information of the target UAV and the countermeasure time requirement. Determine the power advantage ΔP = γ(τ0, δτ) required for the countermeasure signal to successfully control the target UAV under this condition according to the fitting method, where: c is the speed of light, f CA is the C / A code rate, and r is the distance between the device and the trapping position.

[0049] The specific fitting method is as follows: Change the values of (δτ, τ0) through MATLAB simulation, and record the minimum power advantage ΔP required for the countermeasure signal to successfully control the UAV; after generating a large number of data sets, divide them into 10 equal parts on average, take 8 of them as the training set and use the ensemble bagging regression tree (EBRT) model for learning, and use the remaining 2 parts of the data as the test set to test the network. After obtaining the mapping relationship between ΔP and (δτ, τ0), subsequent calculations can be carried out.

[0050] 3.2 Determine the total power of the countermeasure signal for each visible satellite channel emitted by the device according to the fitted power advantage ΔP, the distance L between the device and the target UAV, and the gain of the directional transmitting antenna: where: P th is the total power of each visible satellite channel, λ is the satellite navigation wavelength, G T is the direction gain of the device's directional transmitting antenna, ΔP is the power advantage fitted according to step 3.1, L is the distance between the device and the target UAV, and R is the distance between the device and the trapping position.

[0051] 3.3 Allocate the power of the countermeasure signal for each channel, specifically: Arrange the arrival power of the real visible satellites calculated in step 1 for each channel from small to large according to their power levels P = [P 1 , P 2 , … P i … P N , and the power distribution coefficient for each channel is k i = P i / P N . Ensure that the hijacking signal corresponding to the satellite with a lower real signal power is also allocated a lower power, and the hijacking signal corresponding to the satellite with a higher real signal power is also allocated a higher power.

[0052] Step 4: Select the signal correction loop through the signal selection unit to correct the frequency parameters of the countermeasure signal generated by the device so that it is strictly synchronized with the real signal. Then transmit the countermeasure signal generated by the device to the signal transmitting module, and after upconversion, transmit it to the target UAV.

[0053] After specific experiments, using Spirent GSS8000 satellite navigation signal simulator, the above-mentioned low-altitude economy UAV route safety guarantee system, and two Ublox receivers, according to Figure 6 the wiring method for connection, the physical object of the experimental device is as shown in Figure 7 . The geodetic coordinates of the static position set by the satellite navigation signal simulator are [0.0° 0.0° 0.0m]. The positioning position of Ublox receiver 1 without countermeasure signal access is as shown in Figure 8 (a). The geodetic coordinates of the target point set by the UAV countermeasure device are [0.01° 0.01° 5.0m], which is about 1500m away from the static position set by the satellite navigation signal simulator. The countermeasure signal is sent at a power of -80dBm. The positioning position of Ublox receiver 2 with countermeasure signal access is as shown in Figure 8 (b). It can be seen that Ublox receiver 2 successfully locates the coordinates of the target point, proving that the device successfully sends the countermeasure signal. It can be predicted that the same effect will be achieved if the countermeasure signal is sent to the target UAV.

[0054] Compared with the prior art, the present device can not only quickly detect interference signals, but also take corresponding countermeasure measures after detecting maliciously attacking UAVs and safely lure them to the target position. It is more practical and economical than the devices on the market that only include interference detection functions or countermeasure functions.

[0055] Those skilled in the art can make partial adjustments to the above specific implementation in different ways without departing from the principles and purposes of the present invention. The protection scope of the present invention is subject to the claims and is not limited by the above specific implementation. All implementation solutions within its scope are subject to the present invention.

Claims

1. An air route safety guarantee system for low-altitude economic drones, characterized in that, Comprising: A signal receiving module, a data processing module, a signal transmitting module, and a loop correction module, wherein: the signal receiving module receives real satellite navigation signals, and after down-conversion, outputs them to the data processing module. The data processing module monitors satellite navigation interference in the down-converted signals in real time and generates satellite navigation countermeasure signals to control the target unmanned aerial vehicle (UAV). The signal transmitting module performs digital-to-analog conversion and up-conversion processing on the satellite navigation countermeasure signals and sends them to the target UAV. The loop correction module corrects the parameters of the satellite navigation countermeasure signals through signal sorting to keep them synchronized with the real signals.

2. The low-altitude economy drone route safety guarantee system according to claim 1, characterized in that, The data processing module described above comprises: an external interference signal detection unit and a navigation interference countermeasure signal generation unit, wherein: the external interference signal detection unit performs warm-up acquisition on the satellite signals received by the signal receiving module by grouping them according to orbital planes, and based on the relevant peak morphology and AGC value during the acquisition stage, realizes rapid detection of interference signals; the navigation interference countermeasure signal generation unit predicts key parameters such as the transmission time and Doppler frequency of the countermeasure signals according to the received real signals and information such as the position of the target UAV, and after modifying the data code of the real signals, modulates and generates the baseband signal of the countermeasure signals.

3. The low-altitude economy drone route safety guarantee system according to claim 1, characterized in that The described loop correction module includes: a loop correction unit and a signal selection unit, where: the loop correction unit controls the signal selection unit to select a signal loop for signal self-circulation measurement to obtain an internal delay τ D and feeds it back to the data processing module; the signal selection unit processes the countermeasure signal generation loop or the signal correction loop according to the instruction of the loop correction unit.

4. A method for interference detection for the route safety of low-altitude economic drones, characterized in that, Comprising: Step 1: Group the satellites according to their orbital planes. GPS satellites are divided into 6 groups according to orbital planes ABCDEF, with the number of satellites being 4, 6, 5, 6, 6, and 5 respectively. BDS satellites are divided into 3 groups according to GEO, IGSO, and MEO orbital planes, with the number of satellites being 7, 10, and 27 respectively. Step 2: Perform warm-up acquisition on all satellites. If the detection system is a single GPS or BDS system, all satellites are cyclically detected 3 times. If the detection system is a GPS / BDS combined system, GPS satellites are cyclically detected 2 times in the order of GPS, BDS, GPS, and BDS satellites are cyclically detected 1 time. Step 3: After the warm-up is completed, establish a satellite acquisition status list according to the warm-up results. The GPS system is established in the order of orbital planes A - F, and BDS is established in the order of GEO, IGSO, and MEO. For the satellites captured during the warm-up process, mark their satellite capture identifier as 1 and the counter as 5. For the satellites not captured, mark the satellite capture identifier as 0 and the counter as 0. Step 4: Continuously acquire. Each round of acquisition is carried out again according to a certain number of captured satellites and a certain number of uncaptured satellites, and the acquisition results are divided into two cases, and the value of the counter is updated respectively. Step 5: Perform double-peak detection on the captured satellites. As long as there are double peaks among the captured satellites, it is considered that there is interference; at the same time, detect the AGC value. If it is abnormal, it is considered that there is interference.

5. A navigation countermeasure signal generation method for low-altitude economic UAV route safety based on the system described in any one of claims 1-3, characterized in that, Comprising: Step 1: The signal receiving module receives the capture data of the RF front end and performs tracking demodulation to obtain the signal parameters and navigation message of the currently visible satellites. Step 2: The navigation interference countermeasure signal generation unit constructs countermeasure signals that maintain a high degree of time-frequency synchronization with the real signals according to the signal transmission time, Doppler frequency of the currently visible satellites, and the estimated position and speed information of the target UAV. Step 3: Use precise control of UAV electromagnetic interference countermeasure technology to determine the power level required for the generated countermeasure signal; Step 4: Select a loop correction unit through a signal selection unit to correct the parameters of the countermeasure signal generated by the device to make it strictly synchronized with the real signal. Then transmit the countermeasure signal generated by the device to the signal transmission module, and after up-conversion, transmit it to the target UAV.

6. The navigation countermeasure signal generation method for low-altitude economy UAV route safety according to claim 5, Its feature is that The said step 2 specifically includes: 2.1 Calculate the emission time and Doppler frequency parameters corresponding to the countermeasure signal according to the position of the countermeasure device and the target drone, specifically: emission time in: is the launch time of the real visible satellite, is the pseudorange between the ith visible satellite and the device, is the pseudo-range between the ith visible satellite and the trapping target point, δt a is the local clock error of the device, δt b is the local clock error at the target point, L is the distance between the device and the target UAV, c is the speed of light, τ D is the inherent delay of the device, which is compensated by the loop correction module; Doppler frequency parameter in: is the visible satellite velocity, The speed required to capture the target point, is the unit vector between the visible satellite and the trapping target point, is the satellite clock drift, λ is the wavelength of the satellite navigation signal; 2.2 Update the second information within the week of the navigation message according to the transmission time obtained in step 2.1, and then generate a new check code according to the navigation message coding rule and the updated information, and re-encode the message of the countermeasure signal; 2.3 Calculate the spreading code phase, data code phase and carrier phase of each sampling point in turn, and calculate the value of each sampling point according to the signal structure to complete the navigation message and carrier modulation to generate a digital baseband signal.

7. The navigation countermeasure signal generation method for low-altitude economy UAV route safety according to claim 5, characterized in that, The said step 3 specifically includes: 3.1 Use the step-by-step bias method to achieve stealthy entrapment of the target UAV. Determine the phase offset speed δτ of the countermeasure signal code and the initial code phase difference between the countermeasure signal and the true signal according to the position and speed information of the target UAV and the countermeasure time requirement. Determine the power advantage ΔP = γ(τ0, δτ) required for the countermeasure signal to successfully control the target UAV under this condition according to the fitting method, where: c is the speed of light, f CA is the C / A code rate, and r is the distance between the device and the entrapment position. 3.2 Determine the total power of the countermeasure signal emitted by the device for each visible satellite channel based on the fitted power advantage ΔP, the distance L between the device and the target UAV, and the gain of the directional transmitting antenna: Where: P th is the total power of each visible satellite channel, λ is the satellite navigation wavelength, G T is the directional gain of the device's directional transmitting antenna, ΔP is the power advantage fitted according to step 3.1, L is the distance between the device and the target UAV, and r is the distance between the device and the trapping position; 3.3 Distribute the countermeasure signal power for each channel, specifically: Arrange the arrival power of the truly visible satellites in each channel calculated in step 1 from smallest to largest according to their power levels as P = [P 1 , P 2 , … P i … P N , and the power distribution coefficient for each channel is k i = P i / P N , ensuring that the hijacking signal corresponding to the satellite with a lower true signal power is also allocated a lower power, and the hijacking signal corresponding to the satellite with a higher true signal power is also allocated a higher power.

8. The navigation countermeasure signal generation method for low-altitude economic UAV route safety according to claim 7, characterized in that The said fitting method is specifically as follows: Change the values of (δτ, τ0) through simulation, and record the minimum power advantage ΔP required for the countermeasure signal to successfully control the UAV; after generating a large number of data sets, divide them into 10 equal parts on average, take 8 of them as the training set and use the ensemble bagging regression tree (EBRT) model for learning, and use the remaining 2 parts of the data as the test set to test the network. After obtaining the mapping relationship between ΔP and (δτ, τ0), subsequent calculations can be carried out.

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