Unmanned aerial vehicle low-altitude height observation and automatic terrain following system
By fusing altitude information from laser rangefinders, barometers, and accelerometers, and combining this with altitude controllers and filter optimization, the problem of large altitude estimation errors during low-altitude flight of UAVs was solved, enabling stable control of automatic terrain following and altitude-holding flight.
Patent Information
- Application Number
- CN202511042151.8
- Authority / Receiving Office
- CN · China
- Patent Type
- Applications(China)
- Current Assignee / Owner
- Filing Date
- 2025-07-28
- Publication Date
- 2025-11-11
AI Technical Summary
When existing drones fly at low altitudes, their GPS signals are easily affected by obstacles, resulting in large altitude estimation errors and rapid self-drift, making it impossible to achieve automatic terrain following and obstacle crossing, especially in indoor or complex terrain environments where the accuracy is insufficient.
Altitude information is fused using a laser rangefinder, barometer, and accelerometer. Combined with a PID, MPC, or ARC altitude controller, automatic terrain following and altitude hold are achieved through a mode switching switch. High-pass and low-pass filters are used to optimize sensor data, and altitude observation is performed in conjunction with inertial navigation components.
It improves the accuracy of low-altitude altitude estimation for UAVs, reduces the risk of collisions, enables stable flight in complex terrain, meets different flight needs of operators, and is low in cost and easy to implement in engineering.
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Figure CN120928843A_ABST
Abstract
Description
Technical Field
[0001] This invention belongs to the field of unmanned aerial vehicle (UAV) flight control technology, specifically a UAV low-altitude observation and automatic terrain following system. Background Technology
[0002] With the development of chip technology, more and more existing drones are adopting multi-sensor information fusion to observe and estimate their position information. These sensors include GPS, barometers, inertial navigation units (IMUs), laser rangefinders, binocular cameras, and ultrasonic sensors. Multi-sensor information fusion technology has become a pressing issue to be addressed. Given the same sensor quality, the efficiency and accuracy of the multi-sensor information fusion algorithm will play a decisive role in the observation and estimation of drone position information, and the accuracy of position estimation will determine the drone's position control accuracy. Existing common position observation methods include complementary filtering, Kalman filtering, and moving sliding window estimation. With the increasing computing power of chips, the supported sensor sampling rates are getting higher and higher. High-performance laser rangefinders and barometric altimeters have sampling rates exceeding 50 Hz, significantly improving the computational accuracy of position observation algorithms. This invention uses barometers, accelerometers, and laser rangefinders as signal sources to observe altitude information during low-altitude drone flight. This improves altitude observation accuracy while also solving obstacle crossing problems in low-altitude automatic terrain following, reducing the probability of collisions in complex terrain and ensuring drone flight safety.
[0003] Civilian drones are generally equipped with GPS and barometric altimeters, and can fly at altitudes of tens of kilometers. At high altitudes, drones typically rely on GPS and barometric altimeter data for altitude control, achieving closed-loop control. However, at lower altitudes (below 10 meters), GPS signals are easily affected by buildings or obstacles, resulting in poor signal quality and instability. Therefore, barometric altimeters must be used as the altitude feedback signal. Due to ground airflow backlash and turbulence, barometric altimeter readings are prone to fluctuations and drift at low altitudes, necessitating error compensation and correction from other types of sensors. Laser rangefinders are high-precision distance sensors that, when mounted on the drone's fuselage, can measure altitude. Their built-in light generator eliminates the need for illumination, enabling accurate measurements at night, making them suitable for low-altitude drone altitude measurement. Combining laser and barometric altimeter data allows for high-precision altitude observation, enabling more stable flight indoors or when GPS signals are obstructed by buildings, demonstrating significant potential and value for drone applications.
[0004] For example, a GPS-corrected barometric altitude measurement method, as described in application number "201811458961.1", derives a relationship function between GPS relative altitude and current pressure value, achieving the fusion of UAV altitude data. This method requires no additional auxiliary equipment; relying solely on the existing barometric altimeter and GPS device, it improves the accuracy of altitude data. It avoids GPS sensor jumps during altitude measurement and reduces errors in barometric altimeter measurements, effectively improving altitude measurement accuracy without increasing cost. Furthermore, even if the GPS sensor fails, the method can still perform high-precision altitude measurement based on the relationship function fitted before failure, enhancing the reliability of the measurement system. This method is widely applicable to UAV altitude measurement systems, especially for small UAV platforms where installing large sensors is inconvenient. This method has significant practical implications and broad application prospects for UAV altitude measurement, and it plays a guiding role in the development of new altitude measurement methods. Figure 1 As shown.
[0005] For example, application number "201710061088.1" describes a method and system for measuring the altitude of a UAV based on multi-sensor information fusion. The method includes: (1) setting the priority order of various sensors as: Class I sensors, Class II sensors, and Class III sensors; (2) simultaneously acquiring the altitude measurement values of Class I sensors, Class II sensors, and Class III sensors; (3) determining and selecting the preferred sensor; and (4) obtaining the altitude measurement result after the above altitude data fusion is completed. This invention provides a method for measuring the altitude of a UAV based on multi-sensor information fusion, which can provide real-time, accurate, and reliable altitude information for the UAV, facilitating precise UAV altitude control.
[0006] In summary, most existing technical solutions rely on GPS information for altitude estimation and involve the entire flight profile of the UAV. The relevant information for this invention is the complete design flight altitude, without any special design for low-altitude conditions. In the current market, most small UAVs are used in indoor or low-altitude flight scenarios, and the terrain is not completely flat with many complex obstacles. When the low-altitude GPS signal fails, there will be problems such as large altitude estimation errors and rapid self-drift. Moreover, the measured altitude information cannot distinguish between the UAV's true altitude above the ground and the average sea level altitude, making it impossible to achieve automatic terrain following and obstacle crossing for the UAV based on altitude. Summary of the Invention
[0007] The purpose of this invention is to overcome the shortcomings of the prior art and provide a UAV low-altitude observation and automatic terrain following system.
[0008] To achieve the above objectives, the present invention adopts the following technical solution:
[0009] A UAV low-altitude altitude observation and automatic terrain following system includes a UAV body and a flight control circuit board. The flight control circuit board is provided with an altitude sensor group, an altitude controller and an altitude observation module. The altitude sensor group includes a laser rangefinder sensor located at the bottom of the flight control circuit board, a barometer located at the top of the flight control circuit board and an inertial navigation element, the inertial navigation element including an accelerometer.
[0010] The altitude observation module includes an automatic terrain-following altitude observation module and a constant-altitude cruise altitude observation module;
[0011] The altitude controller is one of PID, MPC, and ARC, and is used for altitude control when the UAV is flying at low altitude. It includes a mode switch, which allows the UAV operator to select automatic terrain following mode or altitude hold mode. The two flight modes correspond to the automatic terrain following altitude observation module and the altitude hold cruise altitude observation module, respectively.
[0012] Preferably, the mathematical model of the UAV body is a second-order quadcopter UAV state-space model, specifically:
[0013]
[0014] Where [h,v] are state variables, representing the drone's altitude and vertical velocity, respectively. The input value a is the drone's vertical acceleration, and the output value h is the drone's altitude.
[0015] Preferably, the altitude sensor group adopts a first-order mathematical model, and its transfer function is:
[0016]
[0017] Each sensor obtains its corresponding time constant parameter t based on its own signal transmission characteristics. C(s) is a constant term that is related to the sensor hardware parameters, and s is the frequency domain Laplace operator.
[0018] Preferably, the automatic terrain-following height observation module is configured as follows: the height information is obtained by double integration after high-pass filtering by the accelerometer and then compared with the weighting coefficient Q. acc Multiplication is performed by multiplying the barometer and laser rangefinder sensors after low-pass filtering with response weighting coefficients, finally obtaining the optimal combination of each sensor in the automatic terrain-following height observation module, specifically including:
[0019]
[0020] Q acc +Q baro +Q laser =1 Equation 4;
[0021]
[0022] Among them, Q acc Q baro Q laser G represents the weighting coefficient. LPF For the low-pass filter model, G HPF This is a high-pass filter model, where w is the cutoff frequency and s is the frequency domain Laplace operator.
[0023] Preferably, during the flight of the UAV, the height of obstacles in the low-altitude terrain is:
[0024] h terr =(h baro Q baro +h acc Q acc )-h laser Equation 6;
[0025] Among them, h terr ,h baro ,h acc ,h laser These represent the height of the obstacle, the height of the barometer, the height of the accelerometer, and the height of the laser rangefinder sensor, respectively.
[0026] Preferably, the altitude-keeping cruise observation module is configured as follows:
[0027]
[0028] Among them, K terr Q is the dynamic weighting coefficient. acc Q baro Q laser G represents the weighting coefficient. LPF For the low-pass filter model, G HPF This is a high-pass filter model, where w is the cutoff frequency and s is the frequency domain Laplace operator.
[0029] In summary, due to the adoption of the above technical solution, the beneficial effects of the present invention are:
[0030] In this invention, the low-altitude altitude estimation of the UAV using fusion observations of barometers, accelerometers, and laser rangefinders is more accurate, avoiding the shortcomings of high-frequency disturbances of accelerometers, low-frequency disturbances of barometers, and the fact that laser rangefinders can only measure the true altitude. At the same time, by setting a mode switching switch, different flight needs of the operator can be met, enabling both automatic terrain following and altitude hold flight. Moreover, each module is an airborne sensor, which is small in size and light in weight, does not require external equipment for auxiliary positioning, is easy to implement in engineering, and has low cost. Attached Figure Description
[0031] Figure 1 This is a schematic diagram of the overall structure of the present invention;
[0032] Figure 2 This is a schematic diagram of the control flow of the present invention;
[0033] Figure 3 This invention relates to the low-altitude automatic terrain following mode (Auto) flight effect of the UAV. Figure 1 ;
[0034] Figure 4 This invention relates to the low-altitude automatic terrain following mode (Auto) flight effect of the UAV. Figure 2 .
[0035] Reference numerals: 1. UAV body; 2. Flight control circuit board; 3. Laser rangefinder sensor; 4. Barometer; 5. Inertial navigation element. Detailed Implementation
[0036] The specific embodiments of the present invention are described in detail below.
[0037] The "range" disclosed in this invention is defined by a lower limit and an upper limit. A given range is defined by selecting a lower limit and an upper limit, which define the boundaries of a particular range. Ranges defined in this way can include or exclude endpoints and can be arbitrarily combined; that is, any lower limit can be combined with any upper limit to form a range. For example, if a range of 10–50 is listed for a specific parameter, it is also expected that ranges of 10–40 and 20–50 are also included. Furthermore, if the minimum range values are 1 and 2, and the maximum range values are 3, 4, and 5, then the following ranges are all expected: 1–3, 1–4, 1–5, 2–3, 2–4, and 2–5. In this application, unless otherwise stated, the numerical range "a–b" represents a shortened representation of any combination of real numbers between a and b, where a and b are real numbers. For example, the numerical range "0–5" means that all real numbers between "0–5" have been listed herein; "0–5" is merely a shortened representation of these numerical combinations.
[0038] Unless otherwise specified, all embodiments and optional embodiments of this application can be combined to form new technical solutions.
[0039] Unless otherwise specified, all technical features and optional technical features of this application may be combined to form new technical solutions.
[0040] Unless otherwise specified, all steps in this application may be performed sequentially or randomly, preferably sequentially. For example, the method includes steps (a) and (b), indicating that the method may include steps (a) and (b) performed sequentially, or it may include steps (b) and (a) performed sequentially. For example, the mention that the method may also include step (c) indicates that step (c) may be added to the method in any order. For example, the method may include steps (a), (b), and (c), or it may include steps (a), (c), and (b), or it may include steps (c), (a), and (b), etc.
[0041] Unless otherwise specified, the terms "comprising" and "including" as used in this application can be open-ended or closed-ended. For example, "comprising" and "including" can mean that other components not listed may also be included, or that only the listed components may be included.
[0042] Unless otherwise specified, the reaction will proceed under normal temperature and pressure conditions.
[0043] Unless otherwise specified, all parts or percentages are by weight or by weight percentage.
[0044] In this invention, all the substances used are known substances that can be purchased or synthesized by known methods.
[0045] In this invention, all the devices or equipment used are conventional devices or equipment known in the art and are readily available.
[0046] The following embodiments further illustrate specific implementations of the UAV low-altitude observation and automatic terrain following system of the present invention. The UAV low-altitude observation and automatic terrain following system of the present invention is not limited to the descriptions in the following embodiments.
[0047] Example 1:
[0048] A low-altitude observation and automatic terrain-following system for unmanned aerial vehicles (UAVs), such as Figure 1-2 As shown, the device includes a drone body 1 and a flight control circuit board 2. The flight control circuit board 2 is equipped with an altitude sensor group, an altitude controller and an altitude observation module. The altitude sensor group includes a laser rangefinder 3 located at the bottom of the flight control circuit board 2, a barometer 4 located at the top of the flight control circuit board 2 and an inertial navigation element 5. The inertial navigation element 5 includes an accelerometer.
[0049] The altitude observation module includes an automatic terrain-following altitude observation module and a constant-altitude cruise altitude observation module;
[0050] The altitude controller is one of PID, MPC, or ARC, and it is used for altitude control when the UAV is flying at low altitude. It includes a mode switch, which allows the UAV operator to select between automatic terrain following mode and altitude hold mode. The two flight modes correspond to the automatic terrain following altitude observation module and the altitude hold cruise altitude observation module, respectively.
[0051] Furthermore, the mathematical model of the UAV body 1 is a second-order quadcopter UAV state-space model, specifically:
[0052]
[0053] Where [h,v] are state variables, representing the drone's altitude and vertical velocity, respectively. The input value a is the drone's vertical acceleration, and the output value h is the drone's altitude.
[0054] Furthermore, the altitude sensor array employs a first-order mathematical model, with the following transfer function:
[0055]
[0056] Each sensor obtains its corresponding time constant parameter t based on its own signal transmission characteristics. C(s) is a constant term that is related to the sensor hardware parameters, and s is the frequency domain Laplace operator.
[0057] Furthermore, the automatic terrain-following height observation module is configured as follows: the accelerometer performs high-pass filtering and double integration to obtain height information, which is then compared with the weighting coefficient Q. acc Multiplication is performed by multiplying the barometer and laser rangefinder sensors after low-pass filtering with response weighting coefficients, finally obtaining the optimal combination of each sensor in the automatic terrain-following height observation module, specifically including:
[0058]
[0059] Q acc +Q baro +Q laser =1 Equation 4;
[0060]
[0061] Among them, Q acc Q baro Q laser G represents the weighting coefficient. LPF For the low-pass filter model, G HPF This is a high-pass filter model, where w is the cutoff frequency and s is the frequency domain Laplace operator.
[0062] Furthermore, during the flight of the UAV body 1, the height of obstacles in the low-altitude terrain is:
[0063] h terr =(h baro Q baro +h acc Q acc )-h laser Equation 6;
[0064] Among them, h terr ,h baro ,h acc ,h laser These represent the height of the obstacle, the height of the barometer, the height of the accelerometer, and the height of the laser rangefinder sensor, respectively.
[0065] Furthermore, the altitude observation module for constant-altitude cruise is configured as follows:
[0066]
[0067] Among them, K terr Q is the dynamic weighting coefficient. acc Q baro Q laser G represents the weighting coefficient. LPF For the low-pass filter model, G HPF This is a high-pass filter model, where w is the cutoff frequency and s is the frequency domain Laplace operator.
[0068] Example 2:
[0069] A low-altitude observation and automatic terrain-following system for unmanned aerial vehicles (UAVs) is applied to UAVs equipped with accelerometers, barometers, and laser rangefinders, wherein the sensors are installed on the UAV as follows: Figure 1 As shown, the flight control circuit board is equipped with an onboard accelerometer and a barometer, and the laser rangefinder is located below the flight control board (it is necessary to ensure that the laser emission path is unobstructed when designing the underside of the fuselage so that the laser rangefinder can emit and receive light).
[0070] Furthermore, this system is used for altitude observation and control during low-altitude flight of the UAV, and the control method block diagram is as follows: Figure 2As shown, the mode switch allows the UAV operator to select either the automatic terrain following mode (Auto) or the altitude hold mode (Alt hold), corresponding to the automatic terrain following altitude observation module and the altitude hold cruise altitude observation module, respectively. The UAV altitude controller can select a general closed-loop error control method, such as model-free proportional-integral-derivative control (PID), model predictive control (MPC), adaptive robust control (ARC), active disturbance rejection control (ADRC), etc. The UAV mathematical model is modeled according to the actual engineering situation and the parameters of the controlled UAV. In this embodiment, a second-order quadcopter UAV state space model is selected, as shown in formula (1). Where is the state variable, representing the UAV altitude and the UAV vertical velocity, respectively. The input is the UAV vertical acceleration, and the output is the UAV altitude.
[0071]
[0072] Where [h,v] are state variables, representing the drone's altitude and vertical velocity, respectively. The input value a is the drone's vertical acceleration, and the output value h is the drone's altitude.
[0073] Furthermore, the accelerometer, barometer, and laser rangefinder sensor models adopt first-order mathematical models, and their transfer functions are shown in formula (2):
[0074]
[0075] Among them, each sensor obtains its corresponding time constant parameter t according to its own signal transmission characteristics, C(s) is a constant term, which is related to the sensor hardware parameters, and s is the frequency domain Laplace operator;
[0076] Furthermore, the time constant parameter t in the model varies for each sensor based on its own signal transmission characteristics. C(s) is a constant term related to the sensor hardware parameters, and s is the frequency domain Laplace operator.
[0077] Furthermore, the automatic terrain following height observation module algorithm is shown in formulas (3)-(5). Its basic principle is that the accelerometer performs high-pass filtering and double integration to obtain the height information, which is then compared with the weighting coefficient Q. acc The barometer and laser rangefinder sensors are multiplied by their response weighting coefficients after low-pass filtering. The optimal combination of sensors in the automatic terrain-following height observation module is then obtained, where the weighting coefficient Q... acc Q baro Q laser Each weight coefficient satisfies formula (4). Corresponding adjustments need to be made according to the sensor hardware parameters to achieve the optimal estimation state. The low-pass filter model is G. LPF The high-pass filter model is G. HPF , where w is the cutoff frequency, as shown in formula (5).
[0078]
[0079] Q acc +Q baro +Q laser =1 Equation 4;
[0080]
[0081] Among them, Q acc Q baro Q laser G represents the weighting coefficient. LPF For the low-pass filter model, G HPF This is a high-pass filter model, where w is the cutoff frequency and s is the frequency domain Laplace operator.
[0082] h terr =(h baro Q baro +h acc Q acc )-h laser Equation 6;
[0083] Among them, h terr ,h baro ,h acc ,h laser These represent the height of the obstacle, the height of the barometer, the height of the accelerometer, and the height of the laser rangefinder sensor, respectively.
[0084] Furthermore, the difference between the altitude-holding cruise altitude observation module algorithm and the automatic terrain-following altitude observation module lies in the fact that the weighting coefficient of the laser rangefinder sensor needs to be dynamically adjusted according to the obstacle height during altitude-holding flight mode. When the obstacle height exceeds the set value, the weighting coefficient is reduced proportionally. For example, if the upper limit of the obstacle is set to 10cm, when the actual obstacle height is 1cm / 5cm / 10cm, it will correspond to Q respectively. laser 0.5Q laser The three different weight coefficients, i.e., formula (3) is adjusted to formula (7), where K terr This refers to the dynamic weighting coefficient.
[0085]
[0086] Among them, K terr Q is the dynamic weighting coefficient. acc Q baro Q laser G represents the weighting coefficient. LPF For the low-pass filter model, G HPFThis is a high-pass filter model, where w is the cutoff frequency and s is the frequency domain Laplace operator.
[0087] Furthermore, to demonstrate the actual flight performance of this algorithm, a 10-minute simulation analysis was conducted after establishing the overall model. The flight performance in low-altitude automatic terrain following mode (Auto) is as follows: Figure 3 As shown, the black area represents the drone's automatic terrain-following altitude, demonstrating its ability to achieve high-precision terrain following and stable flight even without GPS sensor information and at low altitudes. This proves that the altitude observer combined with a laser rangefinder can overcome the high-frequency disturbances of the accelerometer and the low-frequency data drift of the barometer.
[0088] Furthermore, to verify the flight performance when the operator selects the altitude hold mode (Alt hold), a simulation was performed under the same conditions in Mode 2. The flight performance in the low-altitude automatic terrain following mode (Auto) was as follows: Figure 4 As shown, after terrain obstacle compensation using a laser rangefinder and barometer, a fixed altitude flight target can be achieved, and the flight altitude is not affected by the terrain height. However, it is worth noting that the observer cannot completely eliminate the influence of terrain height, which is one of the drawbacks of software observers. But when flying at low altitudes, the effect of controlling the altitude error to the centimeter level is far superior to the accuracy that GPS sensors can achieve. Therefore, both modes can achieve better altitude estimation than when using GPS for position observation, thus enabling more stable altitude-holding flight and having strong practical value.
[0089] By adopting the above technical solution:
[0090] The low-altitude altitude estimation of UAVs using fusion observations of barometers, accelerometers, and laser rangefinders is more accurate, avoiding the shortcomings of high-frequency disturbances in accelerometers, low-frequency disturbances in barometers, and the fact that laser rangefinders can only measure the true altitude.
[0091] By setting a mode switch, different flight needs of operators can be met, enabling both automatic terrain following and altitude hold flight.
[0092] All are airborne sensors, small in size and light in weight, requiring no external equipment for positioning assistance, easy to implement in engineering, and low in cost.
[0093] The above description, in conjunction with specific preferred embodiments, provides a further detailed explanation of the present invention. It should not be construed that the specific implementation of the present invention is limited to these descriptions. For those skilled in the art, various simple deductions or substitutions can be made without departing from the concept of the present invention, and all such modifications and substitutions should be considered within the scope of protection of the present invention.
Claims
1. A low-altitude observation and automatic terrain following system for unmanned aerial vehicles (UAVs), characterized in that, The system includes a drone body (1) and a flight control circuit board (2). The flight control circuit board (2) is equipped with an altitude sensor group, an altitude controller and an altitude observation module. The altitude sensor group includes a laser rangefinder (3) located at the bottom of the flight control circuit board (2), a barometer (4) located at the top of the flight control circuit board (2) and an inertial navigation element (5). The inertial navigation element (5) includes an accelerometer. The altitude observation module includes an automatic terrain-following altitude observation module and a constant-altitude cruise altitude observation module; The altitude controller is one of PID, MPC, and ARC, and is used for altitude control when the UAV is flying at low altitude. It includes a mode switch, which allows the UAV operator to select automatic terrain following mode or altitude hold mode. The two flight modes correspond to the automatic terrain following altitude observation module and the altitude hold cruise altitude observation module, respectively.
2. The UAV low-altitude observation and automatic terrain following system as described in claim 1, characterized in that: The mathematical model of the UAV body (1) is a second-order quadcopter UAV state-space model, specifically: Where [h,v] are state variables, representing the drone's altitude and vertical velocity, respectively. The input value a is the drone's vertical acceleration, and the output value h is the drone's altitude.
3. The UAV low-altitude observation and automatic terrain following system as described in claim 1, characterized in that: The altitude sensor array adopts a first-order mathematical model, and its transfer function is: Each sensor obtains its corresponding time constant parameter t based on its own signal transmission characteristics. C(s) is a constant term that is related to the sensor hardware parameters, and s is the frequency domain Laplace operator.
4. The UAV low-altitude observation and automatic terrain following system as described in claim 1, characterized in that, The automatic terrain-following height observation module is configured such that: after high-pass filtering by the accelerometer, the height information is obtained through double integration and compared with the weighting coefficient Q. acc Multiplication is performed by multiplying the barometer and laser rangefinder sensors after low-pass filtering with response weighting coefficients, finally obtaining the optimal combination of each sensor in the automatic terrain-following height observation module, specifically including: Q acc +Q baro +Q laser =1Equation 4; Among them, Q acc Q baro Q laser G represents the weighting coefficient. LPF For the low-pass filter model, G HPF This is a high-pass filter model, where w is the cutoff frequency and s is the frequency domain Laplace operator.
5. The UAV low-altitude observation and automatic terrain following system as described in claim 1, characterized in that: During the flight of the UAV body (1), the height of obstacles in the low-altitude terrain is: h terr =(h baro Q baro +h acc Q acc )-h laser Equation 6; Among them, h terr ,h baro ,h acc ,h laser These represent the height of the obstacle, the height of the barometer, the height of the accelerometer, and the height of the laser rangefinder sensor, respectively.
6. The UAV low-altitude observation and automatic terrain following system as described in claim 1, characterized in that: The altitude observation module for constant altitude cruise is configured as follows: Among them, K terr Q is the dynamic weighting coefficient. acc Q baro Q laser G represents the weighting coefficient. LPF For the low-pass filter model, G HPF This is a high-pass filter model, where w is the cutoff frequency and s is the frequency domain Laplace operator.
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