Dynamic characteristic testing and modeling method for unmanned aerial vehicle actuator based on real installation state
By installing a test system that simulates the aero hinge torque on the drone actuator, the existing test methods are complicated and error-free, and more accurate dynamic characteristics testing and modeling of the actuator are achieved, which improves the reliability of flight control.
Patent Information
- Application Number
- CN202411774200.2
- Authority / Receiving Office
- CN · China
- Patent Type
- Applications(China)
- Current Assignee / Owner
- Filing Date
- 2024-12-05
- Publication Date
- 2025-03-07
- Estimated Expiration
- 2044-12-05
AI Technical Summary
The existing test methods for dynamic characteristics of drone actuators are cumbersome and have large errors, so it is impossible to accurately consider the moment of inertia load, pneumatic hinge moment and nonlinear factors of mechanical transmission between the servo output shaft and the rudder surface.
Using a test system based on the implementation status, including MEMS inertial guide and suction cups and spring structures that simulate pneumatic hinge torque, it directly measures and simulates the dynamic characteristics of the drone actuator. Through the test of the connection between the computer and the controller, the rudder surface deflection angle and torque are feedback in real time.
The accuracy of actuator modeling is improved, the nonlinearity of mechanical transmission and the inertia load and aerodynamic torque in the real state is taken into account, errors are reduced, and the reliability of flight control law design and the confidence of flight simulation results are enhanced.
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Figure CN119568433A_ABST
Abstract
Description
Technical Field
[0001] The present invention relates to the technical field of unmanned aerial vehicles, and in particular to a method for testing and modeling the dynamic characteristics of an actuator of an unmanned aerial vehicle based on a real-installation state. Background Art
[0002] The UAV actuator is one of the key onboard equipment of the UAV. The actuator receives instructions from the flight control computer, outputs mechanical rotation or linear motion, and drives the corresponding UAV rudder deflection. The rudder deflection causes the change of the UAV's aerodynamic force or torque, thereby changing the movement of the UAV and achieving the purpose of controlling the UAV's movement.
[0003] In the prior art, the actuator is composed of a controller, a servo and cables between them. The controller receives motion instructions from the flight control computer, integrates the servo motion and electrical feedback information for information calculation and processing, generates a servo drive signal and outputs it to the servo. The motor in the servo receives the drive signal from the controller and rotates, driving the internal reducer of the servo to move the servo output shaft to the required position, and feeds back information such as the output shaft position, output shaft speed, armature current, etc. to the controller.
[0004] From the time the flight control computer gives the actuator motion command to the deflection of the rudder, the controller, servo, and the transmission mechanism between the servo output shaft and the rudder (referred to as the transmission mechanism) are involved. The rudder of the drone is the driving object of the actuator. The rudder is subjected to aerodynamic force, which generates a torque on the rudder shaft, called the pneumatic hinge torque. The pneumatic hinge torque is transmitted to the servo through the rudder and the transmission mechanism, becoming the torque / force load (referred to as the force load) of the servo output shaft. The rudder, transmission mechanism, and the internal moving mechanical parts of the servo all have mass or moment of inertia, which together constitute the inertial load of the servo output shaft. The force load and inertial load acting on the servo output shaft have a significant impact on the movement of the actuator. In addition, the friction, clearance, nonlinearity of mechanical transmission (the transmission ratio of the rocker arm in the transmission mechanism changes with the rocker arm deflection angle) and other factors in each link of the servo motor, reducer, output shaft, transmission mechanism, and rudder also have a significant impact on the movement of the actuator.
[0005] The input of the actuator is the deflection command of the rudder surface from the flight control computer, and the output is the deflection angle of the rudder surface. The characteristics of the output response to the input are called the dynamic characteristics of the actuator. The key indicator of the dynamic characteristics of the actuator is the bandwidth, and the key indicator of the static characteristics is the maximum output torque. The bandwidth reflects the speed of the actuator movement. The force / torque on the output shaft of the actuator directly affects the speed of the actuator movement. Therefore, the maximum output force / torque of the actuator also indirectly reflects the speed of the actuator movement.
[0006] The bandwidth of the actuator has a great influence on the flight control performance. The smaller the bandwidth, the slower the response to the flight control computer command, resulting in a larger signal transmission phase delay, which can directly lead to flight control oscillation or even divergence. The dynamic characteristics of the actuator must be considered in the flight control law design and flight control simulation, otherwise the designed control law may not be suitable for the actual UAV, resulting in flight control oscillation or even divergence.
[0007] The dynamic characteristics of the actuator can be represented by its mathematical model. The form of the mathematical model is the transfer function of the output to the input. The transfer function of the general actuator is as follows:
[0008]
[0009] Where s is the Laplace operator, N(s) and D(s) are polynomials with respect to s, and the order of N(s) is smaller than that of D(s).
[0010] In view of the importance of the dynamic characteristics of the actuator, the dynamic characteristics of the actuator must be considered when designing the flight control law and flight simulation of the UAV. To this end, it is necessary to establish a mathematical model of the actuator, that is, to determine N(s) and D(s) in the transfer function of equation (1).
[0011] For actuators whose output is an angle, the actuator is usually Figure 1 The system shown is tested and the test data is analyzed to obtain the transfer function of the actuator. Figure 1 In the figure, the dotted box is the actuator, including the controller and the servo, and the controller and the servo are connected by a cable. The servo is fixed on the test bench, and the controller is powered. A mass block of a certain mass is connected to the end of the servo rocker arm to simulate the inertia load of each link from the servo rocker arm to the rudder surface (inclusive). A spring is connected between the mass block and the fixed frame to simulate the pneumatic hinge torque load of the rudder surface. The test computer is connected to the controller by a cable. The test computer outputs the actuator motion command signal to the controller, and the controller feeds back the servo deflection angle to the test computer. The servo deflection angle is measured by the sensor in the servo and transmitted to the controller through the cable.
[0012] The pneumatic hinge torque load is related to the UAV dynamic pressure and rudder surface deflection angle. The pneumatic hinge torque load on the servo is different under different dynamic pressures and rudder surface deflection angles, which directly affects the motion characteristics of the actuator, that is, the transfer function of the actuator is different under different dynamic pressures and rudder surface deflection angles. According to the dynamic pressure and rudder deflection angle to be measured, the spring stiffness is determined by considering the pneumatic hinge torque coefficient of the UAV. The test computer sends a sweep signal to the controller, and receives and records the servo deflection angle fed back by the controller. Afterwards, the output signal (servo deflection angle) and the input command (sweep signal) are numerically analyzed to obtain the transfer function of the actuator.
[0013] For actuators with linear motion output, use Figure 1 A similar test system, except that the servo output shaft moves in a linear manner, and everything else is the same.
[0014] The above actuator modeling method has the following problems:
[0015] (1) The inertia load converted to the servo output shaft from the servo output shaft to the rudder surface (inclusive) and the pneumatic load converted to the servo output shaft from the pneumatic hinge torque of the rudder surface both change with the rudder surface deflection angle. It is necessary to adjust the mass block and spring stiffness according to the rudder surface deflection angle, which is relatively cumbersome and complicated, and the errors of the converted mass block and spring stiffness are relatively large.
[0016] (2) The above method does not take into account the friction, clearance, nonlinearity of mechanical transmission (the transmission ratio of the rocker arm changes with the rocker arm deflection angle) and other factors existing in each link between the servo output shaft and the rudder surface (inclusive), which causes errors in the test results.
[0017] (3) The feedback signal from the servo does not reflect the actual deflection angle of the rudder surface.
[0018] Due to the above problems, the Figure 1 The method of testing and establishing the actuator model is cumbersome and has large errors. Summary of the invention
[0019] The purpose of the present invention is to provide a dynamic characteristics testing and modeling method for UAV actuators based on the actual installation state, which can improve the modeling accuracy of the actuator, ensure the safety of the control performance within the full envelope flight of the UAV / full motion range of the actuator at the actuator model level, and improve the reliability of the flight control law design and the confidence of the flight simulation results.
[0020] To achieve the above-mentioned purpose, a test system is constructed based on the actuator and rudder of a drone in an in-situ installed state, the test system includes a MEMS inertial navigation system fixed on the surface of the rudder and at the position closest to the rudder rotation axis, the MEMS inertial navigation system is connected to a test computer and a test power supply through cables respectively; a suction cup located on the surface of the rudder is installed at a distance L from the MEMS inertial navigation system, the suction cup is connected to a spring through a rope, and the other end of the spring is connected to a fixed fixing frame; when the rudder deflects 0°, the rope at one end of the spring is just in a straight state.
[0021] Preferably, the actuator includes a controller and a servo, the controller is connected to the test computer and the UAV power supply respectively through the cable, and the controller and the servo are also connected through the cable; the rudder surface rotating shaft is connected to a rudder surface rocker arm, the servo output shaft of the servo is connected to a servo rocker arm, and the rudder surface rocker arm and the servo rocker arm are connected through a connecting rod.
[0022] Preferably, the MEMS inertial navigation system is fixed to the rudder surface by double-sided tape.
[0023] Preferably, if the rudder surface is installed horizontally, the spring is installed below the rudder surface; if the rudder surface is installed vertically, the spring is installed on either side of the rudder surface.
[0024] Preferably, the method for determining the size L is specifically as follows:
[0025] Assuming the spring stiffness is k, the derivative of the aerodynamic hinge torque coefficient of the drone rudder is The dynamic pressure of the drone is q, and the aerodynamic hinge torque generated by the deflection of the rudder surface δ is Among them, S is the aerodynamic reference area, l is the length of the drone, when the rudder deflects δ, the distance the suction cup moves up and down or left and right is x = Lδπ / 180, the tension of the spring is F = kx = kLδπ / 180, and the torque on the rudder shaft is M = FL = kL 2 δπ / 180, the torque of the spring on the rudder shaft is the same as the pneumatic hinge torque, that is, M = M j ,have to:
[0026]
[0027] Preferably, the construction of the test system specifically includes the following steps:
[0028] S1. Fix the MEMS inertial navigation system on the rudder and connect all cables;
[0029] S2. Select a spring with a spring stiffness of k, select the maximum dynamic pressure state of the drone within the flight envelope, use the maximum dynamic pressure to calculate L, and determine the installation position of the suction cup;
[0030] S3, the test computer sets the deflection command of the rudder surface to 0, installs the suction cup, rope and spring at a distance L from the rudder surface rotation axis, and adjusts the length of the rope or the position of the fixing frame so that the tension generated by the spring makes the rope just straight;
[0031] S4, the test computer sends a 0° deflection command, and the command step increases by Δ every 5 seconds until the rudder deflects to the specified maximum deflection;
[0032] S5, setting the transfer function structure of the actuator;
[0033] S6. For each collected step response data, a set of parameters (n0, n1, d0, d1, d2) in the set actuator structure is determined using the system identification app of MATLAB so that the step response of the transfer function approaches the actual step response;
[0034] S7. For each set of transfer functions corresponding to the parameters (n0, n1, d0, d1, d2), the frequency characteristic curve of the transfer function is plotted using the frequency characteristic plotting function of Matlab, thereby reading the bandwidth of the transfer function;
[0035] S8. Take a set of parameters with the smallest bandwidth (n0, n1, d0, d1, d2) and substitute them into the transfer function as the final transfer function of the actuator.
[0036] Preferably, in step S4, Δ is 2° to 3°.
[0037] Preferably, in step S5, the structure of the transfer function is set as:
[0038]
[0039] Where s is the Laplace operator, N(s) and D(s) are polynomials about s, the order of N(s) is less than that of D(s), n1 and n0 are the coefficient of the linear term and the constant term of the polynomial N(s), d2, d1, and d0 are the coefficient of the quadratic term, the coefficient of the linear term, and the constant term of the polynomial D(s), respectively.
[0040] Therefore, the beneficial effects of the present invention using the above-mentioned UAV actuator dynamic characteristics testing and modeling method based on the actual installation state are:
[0041] (1) Directly using the actual rudder surface, actuator and the transmission mechanism between them, the complicated and tedious conversion process of the inertia load on the servo output shaft and the pneumatic hinge load is avoided, thus ensuring the accuracy of the test.
[0042] (2) The test takes into account factors such as friction, clearance, and nonlinearity of mechanical transmission in each link between the servo output shaft and the rudder surface (inclusive), and the actual state is used to ensure the accuracy of the test.
[0043] (3) The actual deflection angle of the rudder surface is directly used to ensure the accuracy of the test.
[0044] (4) By adopting the maximum dynamic pressure state in the flight envelope and a unidirectional step deflection in the opposite direction to the torque generated by the aerodynamic hinge torque and the gravity of the control surface on its rotating shaft, a model of the actuator in the most unfavorable state can be obtained. The bandwidth of this model is the smallest, which improves the reliability of the flight control law design and the confidence of the flight simulation results.
[0045] The technical solution of the present invention is further described in detail below through the accompanying drawings and embodiments. BRIEF DESCRIPTION OF THE DRAWINGS
[0046] Figure 1 It is a structural diagram of a conventional actuator modeling test system;
[0047] Figure 2 It is a structural schematic diagram of a test system in a method for testing and modeling the dynamic characteristics of an actuator of a UAV based on an actual installation state of the present invention.
[0048] Reference numerals
[0049] 1. Actuator; 2. Controller; 3. UAV power supply; 4. Servo; 5. Cable; 6. Test computer; 7. Test bench; 8. Servo rocker arm; 9. Mass block; 10. Spring; 11. Fixing bracket; 12. Rudder; 13. MEMS inertial navigation; 14. Test power supply; 15. Rudder shaft; 16. Suction cup; 17. Rope; 18. Rudder rocker arm; 19. Connecting rod. DETAILED DESCRIPTION
[0050] The technical solution of the present invention is further described below through the accompanying drawings and embodiments.
[0051] Unless otherwise defined, the technical terms or scientific terms used in the present invention should be understood by people with ordinary skills in the field to which the present invention belongs. The words "first", "second" and similar words used in the present invention do not indicate any order, quantity or importance, but are only used to distinguish different components. "Include" or "comprise" and similar words mean that the elements or objects appearing before the word include the elements or objects listed after the word and their equivalents, without excluding other elements or objects. "Connect" or "connected" and similar words are not limited to physical or mechanical connections, but may include electrical connections, whether direct or indirect. "Up", "down", "left", "right" and the like are only used to indicate relative positional relationships. When the absolute position of the described object changes, the relative positional relationship may also change accordingly.
[0052] Embodiment 1
[0053] like Figure 2 As shown, the present invention provides a method for testing and modeling the dynamic characteristics of an actuator of a UAV based on a real installation state, including a test system constructed based on the actuator 1 and the rudder 12 of the UAV in an in-situ installation state, that is, the test is carried out when the UAV is parked on the ground, and the rudder 12, the actuator 1, and the mechanical transmission mechanism from the servo output shaft to the rudder 12 are all in a real installation state.
[0054] The actuator 1 includes a controller 2 and a steering gear 4. The controller 2 is connected to a test computer 6 and a UAV power supply 3 through cables 5, and the controller 2 and the steering gear 4 are also connected through cables 5. A rudder rocker arm 18 is connected to the rudder rotating shaft 15, and a rudder rocker arm 8 is connected to the rudder output shaft of the rudder 4. The rudder rocker arm 18 and the rudder rocker arm 8 are connected through a connecting rod 19.
[0055] The test system includes a MEMS inertial navigation system 13 fixed on the surface of the rudder surface 12 and at the closest position to the rudder surface rotating shaft 15, which is used to measure the rudder surface deflection angle. The position of the MEMS inertial navigation system 13 is as close to the rudder surface rotating shaft 15 as possible, so that its influence on the moment of inertia of the rudder surface 12 relative to the rudder surface rotating shaft 15 can be ignored, so as to improve the test accuracy. Since the volume and weight of the MEMS inertial navigation system 13 are very small, in this embodiment, the MEMS inertial navigation system 13 is fixed on the surface of the rudder surface 12 by double-sided tape, and the MEMS inertial navigation system 13 is connected to the test computer 6 by a cable 5, and the rudder surface deflection angle signal is fed back to the test computer 6. The MEMS inertial navigation system 13 is also connected to the test power supply 14 by a cable 5 to obtain working power.
[0056] A suction cup 16 is installed on the surface of the rudder surface 12 at a distance L from the MEMS inertial navigation system 13. The suction cup 16 is connected to a spring 10 through a rope 17. The other end of the spring 10 is connected to a fixed bracket 11. The spring 10 simulates the pneumatic hinge torque on the rudder surface 12. When the rudder surface 12 deflects 0°, the rope 17 at one end of the spring 10 is ensured to be just in a straight state, and the spring 10 has a very small tension. The placement of the rudder surface 12 is different, and the positions of other structures also need to be adjusted accordingly. If the rudder surface 12 is installed horizontally, the spring 10 is installed below the rudder surface 12. If the rudder surface 12 is installed vertically, the spring 10 is installed on either side of the rudder surface 12.
[0057] The method for determining the above-mentioned dimension L is specifically as follows:
[0058] Assuming the spring stiffness is k, the derivative of the aerodynamic hinge torque coefficient of the drone rudder is The dynamic pressure of the drone is q (unit N / m 2 ), then the aerodynamic hinge torque generated by the deflection of the rudder surface δ (unit: °) is Among them, S (unit m 2 ) is the aerodynamic reference area, l (unit: m) is the length of the drone, when the rudder deflects δ, the distance x of the suction cup moving up and down or left and right is = Lδπ / 180, the tension of the spring is F = kx = kLδπ / 180, and the torque on the rudder shaft is M = FL = kL 2 δπ / 180, the torque of the spring on the rudder shaft is the same as the pneumatic hinge torque, that is, M = M j ,have to:
[0059]
[0060] In this embodiment, the dynamic characteristics testing and modeling method of the UAV actuator based on the actual installation state specifically includes the following steps:
[0061] S1, fix the MEMS inertial navigation system 13 on the control surface 12 and connect all cables 5, such as Figure 2 Install and wire as shown.
[0062] S2. Select an appropriate spring 10 with a spring stiffness of k. Select the maximum dynamic pressure state of the UAV within the flight envelope. Use the maximum dynamic pressure to calculate L using formula (2) to determine the installation position of the suction cup 16.
[0063] S3. The test computer 6 gives a rudder surface deflection instruction of 0, installs a suction cup 16, a rope 17, and a spring 10 at a distance L from the rudder surface rotation axis 15, and adjusts the length of the rope 17 or the position of the fixing frame 11 so that the tension generated by the spring 10 makes the rope 17 just straight.
[0064] S4, the test computer 6 sends a 0° deflection command, and the command is subsequently increased by Δ every 5 seconds, Δ is generally 2° to 3°, until the control surface 12 deflects to the specified maximum deflection.
[0065] S5. Set the transfer function structure of the actuator 1. In this embodiment, the transfer function structure is set as:
[0066]
[0067] Where s is the Laplace operator, N(s) and D(s) are polynomials about s, the order of N(s) is less than that of D(s), n1 and n0 are the coefficient of the linear term and the constant term of the polynomial N(s), d2, d1, and d0 are the coefficient of the quadratic term, the coefficient of the linear term, and the constant term of the polynomial D(s), respectively.
[0068] S6. For each collected step response data, use the system identification app of MATLAB to determine a set of parameters (n0, n1, d0, d1, d2) in the set actuator structure so that the step response of the transfer function (3) is close to the actual step response.
[0069] S7. For each set of transfer functions corresponding to parameters (n0, n1, d0, d1, d2), use the frequency characteristic drawing function of MATLAB to draw the frequency characteristic curve of the transfer function, thereby reading the bandwidth of the transfer function.
[0070] S8. Take a set of parameters with the smallest bandwidth (n0, n1, d0, d1, d2) and substitute them into the transfer function formula (3) as the final transfer function of actuator 1.
[0071] Therefore, the present invention adopts the above-mentioned UAV actuator dynamic characteristics testing and modeling method based on the actual installation state, which improves the actuator modeling accuracy and takes into account the minimum bandwidth state of the actuator, so as to design the control law based on the actuator model. It can ensure the safety of the control performance in the full envelope flight of the UAV / the full motion range of the actuator at the actuator model level, thereby improving the reliability of the flight control law design and the confidence of the flight simulation results.
[0072] Finally, it should be noted that the above embodiments are only used to illustrate the technical solution of the present invention rather than to limit it. Although the present invention has been described in detail with reference to the preferred embodiments, those skilled in the art should understand that they can still modify or replace the technical solution of the present invention with equivalents, and these modifications or equivalent replacements cannot cause the modified technical solution to deviate from the spirit and scope of the technical solution of the present invention.
Claims
1. A method for testing and modeling the dynamic characteristics of UAV actuators based on actual installation, characterized by: The invention comprises a test system constructed in an in-situ installation state of an actuator and a rudder of a drone, wherein the test system comprises a MEMS inertial navigation system fixed on the surface of the rudder and at a position closest to a rotation axis of the rudder, wherein the MEMS inertial navigation system is respectively connected to a test computer and a test power supply through cables; a suction cup located on the surface of the rudder is installed at a distance L from the MEMS inertial navigation system, wherein the suction cup is connected to a spring through a rope, and the other end of the spring is connected to a fixed fixing frame; when the rudder is deflected by 0°, the rope at one end of the spring is just in a straightened state.
2. The method for testing and modeling the dynamic characteristics of an unmanned aerial vehicle actuator based on actual installation state according to claim 1 is characterized in that: The actuator includes a controller and a servo, wherein the controller is connected to the test computer and the UAV power supply respectively through the cable, and the controller and the servo are also connected through the cable; a rudder rocker arm is connected to the rudder rotating shaft, a servo rocker arm is connected to the servo output shaft of the servo, and the rudder rocker arm and the servo rocker arm are connected through a connecting rod.
3. The method for testing and modeling the dynamic characteristics of a drone actuator based on actual installation according to claim 1, characterized in that: The MEMS inertial navigation system is fixed on the rudder surface by double-sided adhesive.
4. The method for testing and modeling the dynamic characteristics of a drone actuator based on actual installation according to claim 1, characterized in that: If the rudder surface is installed horizontally, the spring is installed below the rudder surface; if the rudder surface is installed vertically, the spring is installed on either side of the rudder surface.
5. The method for testing and modeling the dynamic characteristics of an unmanned aerial vehicle actuator based on actual installation state according to claim 1 is characterized in that: The specific method for determining the dimension L is: Assuming the spring stiffness is k, the derivative of the aerodynamic hinge torque coefficient of the drone rudder is The dynamic pressure of the drone is q, and the aerodynamic hinge torque generated by the deflection of the rudder surface δ is Among them, S is the aerodynamic reference area, l is the length of the drone, when the rudder deflects δ, the distance the suction cup moves up and down or left and right is x = Lδπ / 180, the tension of the spring is F = kx = kLδπ / 180, and the torque on the rudder shaft is M = FL = kL 2 δπ / 180, the torque of the spring on the rudder shaft is the same as the pneumatic hinge torque, that is, M = M j ,have to:
6. The method for testing and modeling the dynamic characteristics of a drone actuator based on actual installation state according to claim 1 is characterized in that: The construction of the test system specifically includes the following steps: S1. Fix the MEMS inertial navigation system on the rudder and connect all cables; S2. Select a spring with a spring stiffness of k, select the maximum dynamic pressure state of the drone within the flight envelope, use the maximum dynamic pressure to calculate L, and determine the installation position of the suction cup; S3, the test computer sets the deflection command of the rudder surface to 0, installs the suction cup, rope and spring at a distance L from the rudder surface rotation axis, and adjusts the length of the rope or the position of the fixing frame so that the tension generated by the spring makes the rope just straight; S4, the test computer sends a 0° deflection command, and the command step increases by Δ every 5 seconds until the rudder deflects to the specified maximum deflection; S5, setting the transfer function structure of the actuator; S6. For each collected step response data, a set of parameters (n0, n1, d0, d1, d2) in the set actuator structure is determined using the system identification app of MATLAB so that the step response of the transfer function approaches the actual step response; S7. For each set of transfer functions corresponding to the parameters (n0, n1, d0, d1, d2), the frequency characteristic curve of the transfer function is plotted using the frequency characteristic plotting function of Matlab, thereby reading the bandwidth of the transfer function; S8. Take a set of parameters with the smallest bandwidth (n0, n1, d0, d1, d2) and substitute them into the transfer function as the final transfer function of the actuator.
7. The method for testing and modeling the dynamic characteristics of a drone actuator based on actual installation according to claim 6 is characterized in that: In step S4, Δ is set to 2° to 3°.
8. The method for testing and modeling the dynamic characteristics of a drone actuator based on actual installation according to claim 6, characterized in that: In step S5, the structure of the transfer function is set as: Where s is the Laplace operator, N(s) and D(s) are polynomials about s, the order of N(s) is less than that of D(s), n1 and n0 are the coefficient of the linear term and the constant term of the polynomial N(s), d2, d1, and d0 are the coefficient of the quadratic term, the coefficient of the linear term, and the constant term of the polynomial D(s), respectively.
Citation Information
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