Design method of three-dimensional robust guidance law for aircraft based on rotating line-of-sight coordinate system

Abstract

This invention belongs to the field of aircraft homing guidance technology, and relates to a design method for a three-dimensional robust guidance law for aircraft based on a rotating line-of-sight coordinate system. Addressing the shortcomings of traditional three-dimensional proportional guidance laws in providing insufficient guidance accuracy for highly maneuverable targets and the complex channel coupling processing in existing three-dimensional robust guidance methods, this invention transforms the three-dimensional guidance problem into a two-dimensional robust guidance problem within a plane based on the concept of a guidance plane in a rotating line-of-sight coordinate system, thus simplifying the design complexity. Two types of finite-time guidance methods are proposed: one is a finite-time input-to-state stability guidance method combining fractional feedback control and high-gain proportional control; the other is a finite-time composite guidance method based on a fixed-time convergent extended state observer and a generalized super-torsional perturbation observer. Lyapunov analysis verifies that the proposed method exhibits good stability, effectively improving guidance accuracy for highly maneuverable targets while balancing guidance robustness and engineering feasibility.

Description

Technical Field

[0001] This invention belongs to the field of aircraft homing guidance technology, and relates to a design method for a three-dimensional robust guidance law for aircraft based on a rotating line-of-sight coordinate system. Background Technology

[0002] Three-dimensional robust guidance technology enables precise guidance of aircraft to targets under uncertain disturbances such as high-G maneuvers. Traditional three-dimensional proportional guidance laws are simple, easy to implement, and possess a certain degree of robustness to noise. However, when facing highly maneuverable targets, proportional guidance laws struggle to achieve precise guidance without speed and G-force advantages. Therefore, various advanced control methods have been introduced to address target maneuvering disturbances, such as... Control, sliding mode control, input-to-state stable control, and observer-based composite control, etc.

[0003] Most three-dimensional robust guidance methods are derived based on the idea of ​​decoupling guidance channels. A common approach is to use equivalent control terms to cancel out coupling terms between channels, thereby extending planar robust guidance methods to the three-dimensional case, such as three-dimensional adaptive sliding mode guidance methods and composite guidance methods based on perturbation observers. However, equivalent control terms not only increase the complexity of guidance commands but may also lead to higher fuel consumption. Another approach is to eliminate the effects of cross-coupling during the guidance method design process by selecting an appropriate Lyapunov function; this method has been implemented in various guidance methods. Although this design method avoids the effects of coupling terms, its design process is relatively complex and conservative, and the generated dual-channel guidance commands may still lead to fuel waste and may require more computational resources.

[0004] Existing related patent technologies also have certain limitations:

[0005] The patent "An Optimal Guidance Method Under Weak Overload Capacity Constraints" (CN120178899A) proposes a guidance method based on optimal control under weak overload capacity constraints. Based on solving for the optimal guidance method in the relative motion coordinate system, it establishes an optimal control problem and designs a variable-coefficient bias term that considers overload capacity limitations to achieve precise guidance of highly maneuverable targets. However, optimal control guidance methods often require significant computational resources and time, making them difficult to implement in practical applications.

[0006] The patent "An Adaptive Fast Sliding Mode Guidance Method for Air-to-Air Missiles in a Simulated Environment" (CN117590859A) proposes a fixed-time non-singular fast terminal sliding mode guidance law. Based on a three-dimensional guidance model, line-of-sight normal pitch guidance law and line-of-sight normal yaw guidance law are designed respectively. This method ignores the coupling terms of the pitch and yaw channels, but increases the complexity of guidance commands and fuel consumption.

[0007] In summary, existing three-dimensional robust guidance technologies still face many unresolved issues: traditional proportional guidance laws rely on acceleration advantages when dealing with highly maneuverable targets, making it difficult to guarantee guidance accuracy; various advanced control methods introduce complex equivalent control terms, increasing fuel consumption and command complexity. Therefore, there is an urgent need to explore a more efficient three-dimensional robust guidance design paradigm. Summary of the Invention

[0008] To address the aforementioned problems, this invention provides a three-dimensional robust guidance law design method for aircraft based on a rotating line-of-sight coordinate system, simplifying the three-dimensional robust guidance law design problem. Based on the fundamental principles of vector operations, the three-dimensional guidance problem can be transformed into a two-dimensional guidance problem on the guidance plane in a rotating line-of-sight coordinate system. Compared to the traditional three-dimensional guidance law design process, the design parameters required for the two-dimensional guidance law are reduced by half. As seen in the finite-time input to state-stable guidance law design case, only a set of fractional-order feedback control gains and proportional control coefficients are needed to meet the design requirements. Similarly, as seen in the disturbance observation compensation guidance law design case, only one observer is needed to meet the design requirements. Lyapunov analysis and numerical simulation analysis verify the feasibility of the proposed guidance law design method. In this invention, by transforming the three-dimensional guidance problem into a two-dimensional problem on the guidance plane in a rotating line-of-sight coordinate system, the design complexity is reduced without altering the design accuracy of the guidance law.

[0009] The technical solution of the present invention:

[0010] A method for designing a three-dimensional robust guidance law for an aircraft based on a rotating line-of-sight coordinate system, comprising the following steps:

[0011] Step 1: Construction of the basic guidance model and terminal guidance method design model;

[0012] In three-dimensional space, a mathematical model of the relative motion between the aircraft and the target and a design model of the guidance method are established. Then, based on the basic principle of vector operation, a rotating line-of-sight coordinate system is established. Based on this coordinate system, the three-dimensional guidance problem can be transformed into a two-dimensional guidance problem, which reduces the design complexity of the guidance system. In step 2, a specific guidance method is designed for the two-dimensional guidance model.

[0013] Step 1.1, Basic Guidance Model;

[0014] Based on the principles of kinematics, a mathematical model of the relative motion between the aircraft and the target can be established as the basis for the simplification of the guidance model in the subsequent step 1.2; where Equation (1) is the kinematic equation and Equation (2) is the dynamic equation.

[0015] (1)

[0016] (2)

[0017] In the formula: superscript " " represents the first derivative; The relative distance vector between the aircraft and the target. ; The relative distance between the aircraft and the target; The velocity vector of the target; The velocity vector of the aircraft; The rotational angular velocity vector in the line-of-sight coordinate system. ; , , Representing the line-of-sight coordinate system , , The unit vector of the axis; and These are the elevation angle and the azimuth angle of the line of sight, respectively. and These are the elevation angle and azimuth rate of the line of sight, respectively. Let be the relative velocity vector between the aircraft and the target. ; The acceleration vector of the target; This is the acceleration vector of the aircraft.

[0018] Based on equations (1) and (2) of the mathematical model of the relative motion between the aircraft and the target, the design model of the three-dimensional guidance method is derived as follows:

[0019] (3)

[0020] In the formula: superscript " " denotes the second derivative; , , The target acceleration vector in the line-of-sight coordinate system , , The acceleration components of the axis; , , The acceleration vectors of the aircraft in the line-of-sight coordinate system are respectively , , The acceleration components of the axis.

[0021] Step 1.2: Establish a rotating viewpoint coordinate system;

[0022] Based on the mathematical model of the relative motion between the aircraft and the target established in step 1.1, and based on the fundamental principles of vector operations, a rotating line-of-sight coordinate system is constructed. This coordinate system can transform the three-dimensional guidance model into a two-dimensional guidance model, reducing the design complexity of the guidance method. The axes of the rotating line-of-sight coordinate system... , and Defined by the following unit vector:

[0023] (4)

[0024] In the formula: , , Representing the rotating view coordinate system , , The unit vector of the axis; Angular velocity vector exist shaft and Sum of the components of the axis, , for The modulus, Therefore, the rotational angular velocity vector of the line-of-sight coordinate system can be expressed as:

[0025] (5)

[0026] In the formula: It is the angular velocity vector of the guidance plane. Angular velocity vector exist The magnitude of the axis components. Rotation of the line-of-sight coordinate system. , and The equation of motion for the axis can be derived as follows:

[0027] (6)

[0028] Step 1.3, Terminal Guidance Method Design Model;

[0029] Based on the three-dimensional guidance method design model established in step 1.1 and the rotating line-of-sight coordinate system constructed in step 1.2, the three-dimensional guidance problem is transformed into a two-dimensional guidance problem through the basic principle of vector operation. The three-dimensional guidance method design model shown in equation (3) can be derived as follows:

[0030] (7)

[0031] The guidance method design model shown in Equation (7) is derived from the coordinate system through rotation. Planar guidance can be achieved by zeroing the line-of-sight angular rate. Only the line-of-sight angular rate needs to be considered in the guidance model. The relevant formula is the third formula in equation (7), which is the design model of the robust guidance method in a two-dimensional plane. Equation (7) can be further derived into the design model of the terminal guidance method:

[0032] (8)

[0033] In the formula: The acceleration of the target in the plane; This is the acceleration of the aircraft in the plane.

[0034] The terminal guidance design model shown in Equation (8) is used for the design of the finite-time input to the state-stable guidance method and the composite finite-time guidance method in subsequent step 2. To design the observer, the guidance model shown in Equation (8) is further modified, defining... , The terminal guidance method design model shown in equation (8) can be derived as follows:

[0035] (9)

[0036] The terminal guidance method design models shown in Equation (9) and Equation (8) are only different in form. Equation (9) can be used for observer design and is used as an observer design model.

[0037] Step 2: Two-dimensional robust guidance method;

[0038] Based on the terminal guidance method design model shown in equation (8) established in step 1, a finite-time input-to-state stable guidance method and a composite finite-time convergent guidance method are designed. Both guidance methods can achieve precise guidance for high-speed maneuvering targets. In step 2.1, a finite-time input-to-state stable guidance method is designed based on equation (8). In step 2.2, based on step 2.1, a fixed-time convergent expanding state observer and a generalized superspiral perturbation observer are introduced to observe the maneuvering characteristics of the target, and a composite finite-time convergent guidance method is designed based on equation (9).

[0039] Step 2.1: Finite-time input to state stability guidance method;

[0040] Based on the terminal guidance method design model shown in Equation (8), a robust guidance method similar to proportional guidance is designed:

[0041] (10)

[0042] In the formula: This is the proportionality coefficient. ; The gain coefficient for the nonlinear term. ; For nonlinear terms, ; for Gain, For guidance parameters, ; Define symbols The calculation formula is , Symbols The value within, The sign function is used. The guidance system employing the above guidance method is a finite-time input-to-state stable system.

[0043] Step 2.2: Composite finite-time convergence guidance method;

[0044] The target's maneuver is a typical uncertain disturbance, which can be estimated using an observer. Feedforward compensation is designed based on the observed values ​​to suppress the uncertainty. Based on the observer design model shown in Equation (9), a fixed-time convergent expanding state observer is introduced in step 2.2.1, and a generalized superspiral perturbation observer is introduced in step 2.2.2. Based on these two observers, a composite finite-time convergent guidance method is designed in step 2.2.3. Both observers can observe and compensate for the target's maneuver characteristics.

[0045] Step 2.2.1: Fixed-time convergent expansion state observer

[0046] Based on the observer design model shown in Equation (9), the following fixed-time convergent dilation state observer is designed:

[0047] (11)

[0048] In the formula: This is the observation error; For the observer pair Observed values; For the observer pair The observed value, i.e. ; , and The design parameters for the observer are required. , And satisfy the inequality , .function and Defined as:

[0049] (12)

[0050] In the formula: and These are design parameters, requirements , A fixed-time convergent expanding state observer is used to observe the target's maneuvering acceleration. The observation error... and A small neighborhood that will smoothly converge to zero within a fixed time.

[0051] Step 2.2.2, Generalized Superspiral Perturbation Observer

[0052] While the extended state observer in step 2.2.1 is easy to implement, it cannot provide an accurate estimate of the disturbance. In contrast, the disturbance observer based on sliding mode control can obtain an accurate estimate of the matched disturbance. Based on the observer design model shown in equation (9), the following generalized superspiral disturbance observer is introduced:

[0053] (13)

[0054] In the formula: and These are the design parameters of the observer, requiring... , ,constant , The derivative of the target acceleration, observation error and It will converge to zero within a finite amount of time.

[0055] Step 2.2.3: A composite finite-time convergence guidance method based on perturbation observation compensation

[0056] Based on the obtained target acceleration estimate Based on the terminal guidance method design model shown in Equation (8), a composite finite-time guidance method is designed:

[0057] (14)

[0058] The composite finite-time guidance method shown in Equation (14) is designed based on the terminal guidance method design model in Equation (8). The guidance system using the above guidance method is a system with finite-time input to state stability.

[0059] Step 3: Implementation of the three-dimensional guidance method;

[0060] The use of a rotating line-of-sight coordinate system simplifies the design process of robust guidance methods. The two-dimensional robust guidance method designed in step 2 needs to be transformed into a three-dimensional space for application. According to the definition formula (4) of the rotating line-of-sight coordinate system, the unit vector can be represented by the unit vector of the spherical line-of-sight coordinate system:

[0061] (15)

[0062] Having obtained the aircraft's normal acceleration command within the two-dimensional guidance plane, the acceleration command within the plane needs to be... Projected into three-dimensional space for application. Through vector operations, the designed guidance commands can be... Projected onto the spherical line-of-sight coordinate system:

[0063] (16)

[0064] In the formula: and These are the aircraft in the line-of-sight coordinate system. and The acceleration command for the axis. Furthermore, the projection of the target acceleration onto the line-of-sight normal direction on the guidance plane can be obtained as follows:

[0065] (17)

[0066] In the formula: , and The target is in the line-of-sight coordinate system. , and Axis maneuvering commands.

[0067] The beneficial effects of this invention are:

[0068] This invention establishes a fundamental guidance model and a terminal design model, transforming the three-dimensional guidance problem into a planar problem based on a rotating line-of-sight coordinate system. Building upon this, a finite-time input-to-output stability robust guidance method is proposed, introducing a high-gain feedback control term to suppress target maneuvers. This guidance method is simple in structure and easy to implement. Subsequently, a composite guidance method based on a fixed-time convergent expanding state observer and a generalized superspiral perturbation observer is proposed. The observer can quickly estimate the target's maneuvering acceleration and then perform positive compensation. Compared with traditional three-dimensional composite guidance methods, only one observer dynamics needs to be calculated. Addressing the shortcomings of traditional three-dimensional guidance methods—insufficient robustness, difficulty in adapting to maneuvering targets, and complex modeling—this invention simplifies modeling through a guidance plane and proposes a finite-time robust guidance method, which has broad application prospects. Attached Figure Description

[0069] Figure 1 This is a flowchart of a three-dimensional robust guidance law design method for aircraft based on a rotating line-of-sight coordinate system;

[0070] Figure 2 It is a three-dimensional geometric diagram of the aircraft and the target;

[0071] Figure 3 This is a schematic diagram illustrating the geometric relationship between the rotating viewpoint coordinate system and the spherical viewpoint coordinate system;

[0072] Figure 4 This is a schematic diagram of the process of projecting guidance commands from the rotating line-of-sight coordinate system onto the spherical line-of-sight coordinate system.

[0073] Figure 5 This is a schematic diagram of the line-of-sight angular rate in the rotating line-of-sight coordinate system during the simulation of the finite-time guidance method.

[0074] Figure 6 This is a schematic diagram of the line-of-sight elevation and elevation angular velocities in the spherical line-of-sight coordinate system during the simulation of the finite-time guidance method.

[0075] Figure 7 This is a schematic diagram of the line-of-sight azimuth angular velocity in the spherical line-of-sight coordinate system during the simulation of the finite-time guidance method.

[0076] Figure 8 This is a schematic diagram of acceleration commands in a rotating line-of-sight coordinate system during a simulation of a finite-time guidance method.

[0077] Figure 9 In the simulation of finite-time guidance methods, the spherical line-of-sight coordinate system is used. Axis acceleration command diagram;

[0078] Figure 10 In the simulation of finite-time guidance methods, the spherical line-of-sight coordinate system is used. Axis acceleration command diagram;

[0079] Figure 11 This is a schematic diagram of the line-of-sight angular rate in the rotating line-of-sight coordinate system during the simulation of the composite finite-time guidance method.

[0080] Figure 12 This is a schematic diagram of the line-of-sight elevation and elevation angular velocities in the spherical line-of-sight coordinate system during the simulation of the composite finite-time guidance method.

[0081] Figure 13 This is a schematic diagram of the line-of-sight azimuth angular velocity in the spherical line-of-sight coordinate system during the simulation of the composite finite-time guidance method.

[0082] Figure 14 This is a schematic diagram of acceleration commands in the rotating line-of-sight coordinate system during the simulation of the composite finite-time guidance method.

[0083] Figure 15 In the simulation of the composite finite-time guidance method, the spherical line-of-sight coordinate system is used. Axis acceleration command diagram;

[0084] Figure 16In the simulation of the composite finite-time guidance method, the spherical line-of-sight coordinate system is used. Axis acceleration command diagram;

[0085] Figure 17 These are the observation results from each observer of the composite finite-time guidance method;

[0086] Figure 18 This is a magnified view of the observation results from each observer in the composite finite-time guidance method. Detailed Implementation

[0087] The specific embodiments of the present invention will be further described below with reference to the accompanying drawings and technical solutions.

[0088] First, a mathematical model of the relative motion between the aircraft and the target in three-dimensional space was established. Based on the fundamental principles of vector operations, the three-dimensional guidance problem was transformed into a two-dimensional planar robust guidance problem in a rotating line-of-sight coordinate system. A finite-time input-to-state stability guidance method and a composite finite-time guidance method were designed and proven using Lyapunov stability. Subsequently, the guidance acceleration command was projected from the rotating line-of-sight coordinate system to a spherical line-of-sight coordinate system. Finally, a comparative analysis was conducted with traditional proportional guidance and sliding mode guidance methods.

[0089] The flowchart of the design method for three-dimensional robust guidance law of aircraft based on rotating line-of-sight coordinate system is as follows: Figure 1 As shown, the specific process is as follows:

[0090] Step 1: Construction of the basic guidance model and terminal guidance method design model;

[0091] Based on the fundamental principles of vector operations, a three-dimensional spatial guidance model is transformed into a terminal guidance model in the guidance plane of a rotating line-of-sight coordinate system, which is easier to design and analyze, thus laying a mathematical model foundation for the specific design of subsequent guidance methods.

[0092] Step 1.1, Basic Guidance Model;

[0093] Based on the principles of dynamics, the fundamental equations describing the relative motion between the aircraft and the target are derived, and from these, the traditional guidance design model is extended.

[0094] The geometry of the aircraft and target in three-dimensional space is as follows Figure 2 As shown. Consider the aircraft and target as point masses with constant velocities; the autopilot and seeker dynamics of the aircraft are fast enough to be negligible; the angle of attack of the aircraft is small enough to be negligible. Based on the principles of dynamics, the following equations are derived:

[0095] (1)

[0096] (2)

[0097] Based on equations (1) and (2) of the mathematical model of the relative motion between the aircraft and the target, the design model of the three-dimensional guidance method is derived as follows:

[0098] (3)

[0099] Step 1.2: Establish a rotating viewpoint coordinate system;

[0100] By introducing a rotating line-of-sight coordinate system and describing the guidance problem within this system, based on the fundamental principles of vector operations, the three-dimensional guidance problem is transformed into a two-dimensional guidance problem. The axes of this rotating line-of-sight coordinate system (…) , and The unit vector is defined by the following:

[0101] (4)

[0102] Therefore, the original line-of-sight angular velocity can be expressed as:

[0103] (5)

[0104] Solve for each axis of the rotating viewpoint coordinate system ( , and The equations of motion for ( ) are solved as follows:

[0105] (6)

[0106] In the formula: Rotate the line-of-sight coordinate system The unit vector of the axis, therefore, Substituting the line-of-sight angular velocity equation (5) into equation (6), we get:

[0107] (7)

[0108] Similarly, the rotational viewpoint coordinate system can be obtained. and The equations of motion for are as follows:

[0109] (8)

[0110] (9)

[0111] Therefore, rotate the axis of the line-of-sight coordinate system ( , and The equations of motion for () can be rearranged as follows:

[0112] (10)

[0113] Step 1.3, Terminal Guidance Method Design Model;

[0114] Based on the three-dimensional guidance model in step 1.1 and the rotating line-of-sight coordinate system in step 1.2, the three-dimensional guidance problem is transformed into a two-dimensional planar guidance problem. The relative dynamics in the rotating line-of-sight coordinate system can be derived as follows:

[0115] (11)

[0116] In equation (11), the second equation describes the rotation principle of the guidance plane, while the other two equations represent the relative motion within the guidance plane. Furthermore, the second equation can be decoupled from the other two equations, meaning that the rotation of the guidance plane does not directly affect the relative motion of the line of sight.

[0117] In terminal guidance method research, parallel approach is achieved by zeroing the line-of-sight angular rate. Therefore, the terminal guidance method design model can be derived from equation (11), as shown in the following equation:

[0118] (12)

[0119] To design the disturbance observer, the guidance system design model shown in equation (12) is rewritten as follows:

[0120] (13)

[0121] Step 2: Two-dimensional robust guidance method;

[0122] Based on the two-dimensional planar guidance design model shown in equations (12) and (13) derived in step 1, a finite-time input-to-state stability guidance method and a composite finite-time convergence guidance method are designed.

[0123] Step 2.1: Finite-time input to state stability guidance method;

[0124] Considering the guidance design model shown in equation (12), a robust guidance method similar to proportional guidance is designed:

[0125] (14)

[0126] Define a continuous Lyapunov function ,as follows:

[0127] (15)

[0128] Calculate the derivative of equation (15) with respect to time:

[0129] (16)

[0130] Combining equation (12) from system dynamics and equation (14) from guidance methods, equation (16) can be derived as follows:

[0131] (17)

[0132] when At this point, the seeker will be unable to effectively measure relative motion, and the last valid command will be executed until the simulation ends. To ensure the seeker can effectively measure the minimum relative motion, equation (17) can be rewritten as follows:

[0133] (18)

[0134] In the formula: , , It is about Boundary function It can be concluded that by using the guidance method shown in Equation (16), the terminal guidance system shown in Equation (12) can be made to reach a stable state within a finite time.

[0135] Step 2.2, Composite Finite-Time Convergence Guidance Method

[0136] Step 2.2.1: Fixed-time convergent expansion state observer

[0137] To achieve the observation of the target's maneuvering acceleration, the following continuous fixed-time convergent expanding state observer is proposed based on the observer design model shown in Equation (13):

[0138] (19)

[0139] Step 2.2.2, Generalized Superspiral Perturbation Observer

[0140] A generalized superspiral disturbance observer based on sliding mode control is introduced to accurately estimate the disturbance. The variables in the observer design model shown in equation (13) are assumed to be... , , and All are measurable, and the rate of change of the target acceleration is uniformly bounded, i.e., there exists a constant. Make Established.

[0141] (20)

[0142] In the formula: and It is a positive number. Observation error. and It will converge to zero within a finite amount of time.

[0143] Step 2.2.3: A composite finite-time convergence guidance method based on perturbation observation compensation

[0144] Based on the fixed-time convergent expanding state observer designed in step 2.2.1 and the generalized superspiral perturbation observer designed in step 2.2.2, the estimated value of the target acceleration can be obtained. To design a composite finite-time convergent guidance method based on disturbance observation compensation for the terminal guidance system shown in equation (12):

[0145] (twenty one)

[0146] Choosing equation (15) as the Lyapunov function, calculating its differential with respect to time, and substituting it into the terminal guidance system shown in equation (12), we can obtain:

[0147] (twenty two)

[0148] In the formula: This refers to observation error. Before the observation error reaches equilibrium, the error... It is uniform and bounded. Let The finite convergence time of the observer is represented by a constant. Such that for any satisfying of They all If true, equation (22) can be expressed as:

[0149] (twenty three)

[0150] Equation (23) shows that the state of the terminal guidance system shown in Equation (12) will not diverge to infinity. When the observation error converges to the neighborhood of the origin, It will be bounded by a small constant, that is, there exists a small constant. , so that for any They all This is true. In this case, equation (23) can be derived as:

[0151] (twenty four)

[0152] (25)

[0153] In the formula: ,if If true, then equation (25) can be simplified to:

[0154] (26)

[0155] It can be concluded that by adopting the composite finite-time convergence guidance method shown in Equation (21), the state of the terminal guidance model shown in Equation (12) can be converged to the vicinity of the origin in a finite time.

[0156] Step 3: Implementation of the three-dimensional guidance method;

[0157] The application of a rotating line-of-sight coordinate system has a significant effect on simplifying the design process of robust guidance methods. Figure 3 The geometric relationship between the rotating view coordinate system and the spherical view coordinate system is shown. According to the definition of the rotating view coordinate system, the unit vector shown in equation (4) can be represented by the unit vector of the spherical view coordinate system. According to the definition of each coordinate axis of the rotating view coordinate system in equation (4), the rotating view coordinate system... The coordinate system of the axis and the spherical line of sight The axes coincide, therefore ;Will Substituting into equation (4) yields , Through vector operations The specific process is as follows:

[0158] (27)

[0159] Designed guidance commands The projection onto the spherical viewpoint coordinate system can be performed using the following vector operations, as follows: Figure 4 As shown.

[0160] (28)

[0161] (29)

[0162] Target acceleration In a spherical line-of-sight coordinate system, this is represented as follows:

[0163] (30)

[0164] In the guidance plane, the projection of the target acceleration onto the line-of-sight normal can be obtained by the following formula:

[0165] (31)

[0166] The relative motion dynamics can be obtained from equations (3), (28), (29), and (30). Based on equation (31), the following can be obtained: This allows for the verification of the output of the interference observer.

[0167] Step 4: Comparison of the three-dimensional robust guidance method with other guidance methods;

[0168] Step 4.1: Simulation condition settings;

[0169] The performance of the proposed guidance method will be verified through simulations of different aircraft-target scenarios. The proportional guidance method (PNG) (Equation (32)), augmented proportional guidance method (APNG) (Equation (33)) and sliding mode control (SMC) (Equation (34)) guidance methods will be included for comparison.

[0170] (32)

[0171] (33)

[0172] (34)

[0173] In the established simulation scenario, the reference coordinate system is defined as follows: Figure 2 As shown in Table 1, the initial states of the aircraft and the target are given.

[0174] Table 1 Initial conditions for the aircraft and target

[0175]

[0176] The normal acceleration overload of the aircraft is limited to: The target's acceleration takes the following form:

[0177] (35)

[0178] In the formula: This is the acceleration due to gravity.

[0179] The simulation step size was set to 1 ms. The relative kinematics and dynamic equations were integrated using the fourth-order Runge-Kutta method. The integration calculation for the interference observer was performed using the Euler method. The guidance loop continued to operate until the relative distance was less than 100 m, after which the last valid guidance command would continue to be executed until the simulation ended.

[0180] Step 4.2: Comparison of finite-time guidance methods;

[0181] Finite-time input to state stability guidance method (FTISS) Equation (14), whose parameters are set as follows: , , ,as well as 0.01, 0.005, 0.002.

[0182] Table 2 Figures 5 to 11 The performance of the guidance methods against sinusoidal maneuvering targets is demonstrated. Table 2 shows that all selected guidance methods achieve good guidance accuracy, but the FTISS guidance method performs better, exhibiting superior guidance performance compared to the proportional guidance law. Figures 5 to 8 This indicates that SMC reduces the line-of-sight angular rate. convergence to Within the neighborhood of [the target area]. The accuracy of the FTISS guidance method varies with parameters. Decrease and increase, when At that time, the FTISS guidance method forces the line-of-sight angular rate convergence to Within its neighborhood. Figures 8 to 10 The figure shows that reducing the parameter This will place higher demands on the aircraft's control acceleration capabilities. Figure 9 and Figure 10 Acceleration saturation can be observed, that is, when At that time, the acceleration command of the FTISS guidance method reached the preset upper limit.

[0183] Table 2 Comparison of final relative distance and miss distance

[0184]

[0185] Step 4.3: Comparison of composite guidance methods;

[0186] The Composite Finite-Time Guidance Method (CFTG) (Equation (20)) will be analyzed, with its parameters set as follows: , , The parameters of the extended state observer (ESO) (Equation (19)) are set as follows: , , The parameters of the Generalized Superspiral Disturbance Observer (GSTDO) (Equation (26)) are set as follows: , , Augmented Proportional Guidance (APN) (Equation (33)) and Sliding Mode Control (SMC)-based guidance method (Equation (34)) will be used for comparative analysis. Furthermore, the target maneuver estimate in the APN method is provided by ESO. Simulation results for sinusoidal maneuvering targets are as follows: Figures 11 to 18 As shown in Table 3.

[0187] The results in Table 3 show that rotating the coordinate system can solve complex three-dimensional guidance problems and reduce the design complexity of three-dimensional guidance methods. Figures 11 to 13The results show that the line-of-sight angular velocities of the four guidance methods converge to near the origin within 6 seconds, with the SMC method exhibiting relatively higher convergence accuracy. Guidance methods with observers show significant fluctuations after 6 seconds. Figures 14 to 16 This indicates that the guidance acceleration command is larger in the early and late stages of the simulation, and smaller when the line-of-sight angular rate converges to near the origin.

[0188] Table 3 Comparison of final relative distance and miss distance

[0189]

[0190] Although observer-based composite guidance methods require more computational resources to solve the observer dynamics, observer-based perturbation estimation and compensation can offset the uncertainties caused by target maneuvers as much as possible. Figure 17 and Figure 18 This indicates that both ESO and GSTDO can accurately obtain estimates of the target step maneuver acceleration. For time-varying perturbations, GSTDO shows superior observation results compared to ESO. Nevertheless, the chattering effect still exists. Furthermore, ESO does not require explicit estimation of the perturbation or the boundaries of its continuous derivative.

Claims

1. A method for designing a three-dimensional robust guidance law for an aircraft based on a rotating line-of-sight coordinate system, characterized in that, The steps are as follows: Step 1: Construction of the basic guidance model and terminal guidance method design model; In three-dimensional space, a mathematical model of the relative motion between the aircraft and the target and a design model of the guidance method are established; then, based on the principle of vector operation, a rotating line-of-sight coordinate system is established to transform the three-dimensional guidance problem into a two-dimensional guidance problem. Step 2: Two-dimensional robust guidance method; Based on the terminal guidance method design model established in step 1, a finite-time input-state-stable guidance method and a composite finite-time convergent guidance method are designed. Both guidance methods can achieve precise guidance for high-speed maneuvering targets. Step 3: Implementation of the three-dimensional guidance method; The two-dimensional robust guidance method designed in step 2 is then applied in three-dimensional space. Step 1 is as follows: Step 1.1, Basic Guidance Model; A mathematical model of the relative motion between the aircraft and the target is established based on the principles of kinematics. Equation (1) is the kinematic equation, and Equation (2) is the dynamic equation. （1） （2） In the formula: superscript " " represents the first derivative; The relative distance vector between the aircraft and the target. ; The relative distance between the aircraft and the target; The velocity vector of the target; The velocity vector of the aircraft; The rotational angular velocity vector in the line-of-sight coordinate system. ; , , Representing the line-of-sight coordinate system , , The unit vector of the axis; and These are the elevation angle and the azimuth angle of the line of sight, respectively. and These are the elevation angle and azimuth rate of the line of sight, respectively. Let be the relative velocity vector between the aircraft and the target. ; The acceleration vector of the target; The acceleration vector of the aircraft; Based on equations (1) and (2) of the mathematical model of the relative motion between the aircraft and the target, the design model of the three-dimensional guidance method is derived as follows: （3） In the formula: superscript " " denotes the second derivative; , , The target acceleration vector in the line-of-sight coordinate system , , The acceleration components of the axis; , , The acceleration vectors of the aircraft in the line-of-sight coordinate system are respectively , , The acceleration components of the axis; Step 1.2: Establish a rotating viewpoint coordinate system; Based on the mathematical model of the relative motion between the aircraft and the target established in step 1.1, a rotating line-of-sight coordinate system is constructed based on the principle of vector operations; the axes of the rotating line-of-sight coordinate system... , and Defined by the following unit vector: （4） In the formula: , , Representing the rotating view coordinate system , , The unit vector of the axis; Angular velocity vector exist shaft and Sum of the components of the axis, , for The modulus, Therefore, the rotational angular velocity vector of the line-of-sight coordinate system is expressed as: （5） In the formula: It is the angular velocity vector of the guidance plane. Angular velocity vector exist The magnitude of the axis components; the rotation of the line-of-sight coordinate system. , and The equation of motion for the shaft is derived as follows: （6） Step 1.3, Terminal Guidance Method Design Model; Based on the three-dimensional guidance method design model established in step 1.1 and the rotating line-of-sight coordinate system constructed in step 1.2, the three-dimensional guidance problem is transformed into a two-dimensional guidance problem based on the principle of vector operation. The three-dimensional guidance method design model shown in equation (3) is derived as follows: （7） The guidance method design model shown in Equation (7) is derived from the coordinate system through rotation. Planar guidance is achieved by zeroing the line-of-sight angular rate. Only the line-of-sight angular rate in the guidance model is considered. The relevant formula, namely the third formula in equation (7), is the design model for a robust guidance method in a two-dimensional plane. Equation (7) is further derived into the design model for a terminal guidance method: （8） In the formula: The acceleration of the target in the plane; The acceleration of the aircraft in the plane; The terminal guidance design model shown in Equation (8) is used for the design of the finite-time input to the state-stable guidance method and the composite finite-time guidance method in subsequent step 2; to design the observer, the guidance model shown in Equation (8) is further modified, and the following definition is made: , The design model for the terminal guidance method shown in equation (8) is derived as follows: （9） Equation (9) is used for observer design.

2. The method for designing a three-dimensional robust guidance law for an aircraft based on a rotating line-of-sight coordinate system according to claim 1, characterized in that, Step 2 is as follows: Step 2.1: Finite-time input to state stability guidance method; Based on the terminal guidance method design model shown in equation (8), a robust guidance method is designed: （10） In the formula: This is the proportionality coefficient. ; The gain coefficient for the nonlinear term. ; For nonlinear terms, ; for Gain, For guidance parameters, ; Define symbols The calculation formula is , Symbols The value within, It is a symbolic function; Step 2.2: Composite finite-time convergence guidance method; Step 2.2.1: Fixed-time convergent expansion state observer Based on the observer design model shown in Equation (9), the following fixed-time convergent dilation state observer is designed: （11） In the formula: This is the observation error; For the observer pair Observed values; For the observer pair The observed value, i.e. ; , and The design parameters for the observer are required. , And satisfy the inequality , ;function and Defined as: （12） In the formula: and These are design parameters. , The target's maneuvering acceleration is observed using a fixed-time convergent-expanding state observer, with observation errors... and The neighborhood that smoothly converges to zero within a fixed time. Step 2.2.2, Generalized Superspiral Perturbation Observer Based on the observer design model shown in Equation (9), the following generalized superspiral perturbation observer is introduced: （13） In the formula: and These are the design parameters of the observer, requiring... , ,constant , The derivative of the target acceleration, observation error and It converges to zero within a finite amount of time; Step 2.2.3: A composite finite-time convergence guidance method based on perturbation observation compensation Based on the obtained target acceleration estimate Based on the terminal guidance method design model shown in Equation (8), a composite finite-time guidance method is designed: （14）。 3. The method for designing a three-dimensional robust guidance law for an aircraft based on a rotating line-of-sight coordinate system according to claim 1, characterized in that, Step 3 is as follows: According to the definition formula (4) of the rotating view coordinate system, the unit vector is represented by the unit vector of the spherical view coordinate system: （15） Having obtained the aircraft's normal acceleration command within the two-dimensional guidance plane, the acceleration command within the plane needs to be... Projected into three-dimensional space for application; through vector operations, the designed guidance commands are... Projected onto the spherical line-of-sight coordinate system: （16） In the formula: and These are the aircraft in the line-of-sight coordinate system. and The acceleration command for the axis; furthermore, on the guidance plane, the projection of the target acceleration onto the line-of-sight normal is obtained in the following way: （17） In the formula: , and The target is in the line-of-sight coordinate system. , and Axis maneuvering commands.

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