A robust trajectory optimization method for tiltrotor aircraft based on direct probability integration method

Abstract

A robust trajectory optimization method for a tiltrotor aircraft based on a direct probability integration method belongs to the field of aircraft technology. The method comprises the following steps: first, obtaining a multi-body dynamics model of the tiltrotor aircraft and constructing a deterministic trajectory optimization problem based on the dynamic equations, mechanical constraints, and mission requirements; second, introducing uncertain variables into the deterministic trajectory optimization problem to obtain an uncertain trajectory optimization problem, which is then converted into a robust trajectory optimization problem; third, using the direct probability integration method to perform multiple sampling of the uncertain variables and quantize the random functions in the robust trajectory optimization problem; finally, substituting the converted random functions into the robust trajectory optimization problem to obtain multiple deterministic trajectory optimization problems, and solving them to obtain the statistical characteristics of the tiltrotor aircraft's state variables and control inputs. The method can obtain optimal control variables and trajectories that are both reliable and robust, with considerable accuracy and significant computational efficiency advantages.

Description

Technical Field

[0001] The present invention belongs to the technical field of aircraft and relates to a tiltrotor aircraft robust trajectory optimization method based on a direct probability integration method. Background Art

[0002] In recent years, with the rapid development of drone technology, tilt-rotor drones (UAVs), with their superior vertical take-off and landing capabilities and efficient cruising speeds, have shown broad application prospects in military, industrial, agricultural, and firefighting and rescue applications. Combining the advantages of helicopters and fixed-wing aircraft, they have become a research hotspot in the field. In actual mission scenarios, tilt-rotor aircraft transition between helicopter and fixed-wing modes by tilting their rotors. However, the complex aerodynamic interactions between the rotors and wings during this transition pose significant challenges to flight control and stability. During the transition period, the shift from rotor-to-wing lift generation leads to significant changes in aerodynamic characteristics, compromising flight stability.

[0003] In response to the above problems, many scholars have conducted extensive research and experiments. The results show that the interaction between the wings during the transition phase of the aircraft is the main cause of flight instability and safety issues. Subsequent in-depth research on configuration design and control strategies has basically solved a series of problems existing in the transition phase of tilt-rotor aircraft, and improved flight performance and stability. However, most current optimization studies are based on deterministic environments, ignoring the impact of uncertainties such as external interference or experimental measurement errors that may occur during the flight phase. In other words, trajectory optimization only focuses on improving trajectory performance under nominal conditions, without considering the robustness and reliability of the trajectory under real conditions. This will result in the inability to effectively deal with unknown interference in actual missions and thus fail to achieve the desired goals. It is also accompanied by safety risks. Therefore, it is crucial to implement an effective robust optimization strategy. For example, the Chinese invention patent [CN109976154A] provides an aircraft trajectory optimization method based on chaotic polynomials and sequential convex optimization, which solves the problem of uncertainty analysis and optimization loop nesting in the robust optimization problem of aircraft trajectories, which makes it difficult to solve. However, this method has problems such as low computational efficiency, poor versatility, and difficulty in handling multidimensional uncertainty, which limits its application in complex constraints and strong nonlinear robust optimization problems. To improve computational efficiency, the Chinese invention patent [CN115793438A] proposes a robust trajectory optimization method for aircraft based on the random response surface method and non-embedded chaotic polynomials. This method, based on the idea of ​​a proxy model, obtains a chaotic polynomial model by fitting the relationship between the system output and input. A small amount of sampling combined with a typical sampling method can be used to calculate the polynomial coefficients. Although this method can reduce the model transformation dimension and scale and has high computational efficiency, it expands the state to be optimized by 45 times when solving the robust optimization problem of a gliding trajectory with uncertainty and a random variable dimension of 8. It also requires a series of steps such as convexification and discretization, making the process relatively complex and failing to avoid dimensionality increase for the stochastic optimal control problem. Summary of the Invention

[0004] To address the aforementioned technical issues, the present invention proposes a robust trajectory optimization method for tiltrotor aircraft based on a direct probability integration method. By modeling the tiltrotor aircraft's multibody dynamics and performing relevant constraint analysis, the present invention establishes a robust trajectory optimization problem model based on the mechanical constraints and stability requirements of the aircraft's transition phase, while also considering the uncertainty of aerodynamic parameters during actual operation. Because the problem equation contains random functions that cannot be directly solved, the random functions are quantified using a direct probability integration method, ultimately transforming the problem into a deterministic trajectory optimization problem that can be directly solved.

[0005] In order to achieve the above object, the technical solution adopted by the present invention is:

[0006] A tiltrotor aircraft robust trajectory optimization method based on a direct probability integration method, the tiltrotor aircraft robust trajectory optimization method comprising the following steps:

[0007] Step 1: Consider the tiltrotor aircraft as a tiltrotor aircraft composed of seven parts, and derive a multi-body dynamic model of the tiltrotor aircraft;

[0008] Step 2: Construct a deterministic trajectory optimization problem based on the dynamic equations, mechanical constraints, and mission requirements of the tiltrotor aircraft.

[0009] Step 3: Introduce uncertainty variables into the deterministic trajectory optimization problem to obtain the uncertain trajectory optimization problem, and transform it into a robust trajectory optimization problem;

[0010] Step 4: Use direct probability integration method to calculate the uncertainty variables. Subsampling,quantizes the random function in the robust trajectory optimization problem;

[0011] Step 5: Substitute the transformed random function into the robust trajectory optimization problem to obtain A deterministic trajectory optimization problem is solved to obtain the statistical characteristics of the tiltrotor aircraft state and control input.

[0012] The beneficial effects of the present invention are:

[0013] Starting from the deterministic trajectory optimization problem, the present invention considers probabilistic uncertainty and establishes a formulation for the uncertainty trajectory optimization problem. Then, based on robust optimization theory, a robust trajectory optimization formula is proposed. By constructing an uncertainty quantification propagation model based on the direct probability integration method, the transformation of the uncertainty stochastic problem is achieved, ultimately obtaining a deterministic optimization problem that can be directly solved with considerable accuracy and significant computational efficiency advantages. The constructed uncertainty quantification propagation model based on the direct probability integration method can effectively transform the stochastic problem of dynamic parameter uncertainty through a point selection method based on the GF deviation, without requiring dimensionality increase of the original problem, resulting in higher computational efficiency and greater versatility. BRIEF DESCRIPTION OF THE DRAWINGS

[0014] Figure 1 Flowchart of the present invention.

[0015] Figure 2 Schematic diagram of the rigid body model of the tilt-rotor aircraft in the present invention.

[0016] Figure 3 Schematic diagram of the coordinates of the tilt-rotor aircraft in the present invention.

[0017] Figure 4 Schematic diagram of aerodynamic principles in the present invention.

[0018] Figure 5 Schematic diagram of the multi-body subsystem and constraint relationship of the tiltrotor aircraft in the present invention.

[0019] Figure 6 1 is a control input curve of a tiltrotor aircraft under different disturbance levels in an embodiment of the present invention; Figure 6 (a) is the control input curve of the nacelle inclination angle; Figure 6 (b) in the figure is the control input curve of the rotor speed.

[0020] Figure 7 is the mean trajectory of the tiltrotor aircraft under different disturbance levels in the embodiment of the present invention; Figure 7 (a) is the trajectory mean obtained by integrating the deterministic input; Figure 7 (b) in the figure is the trajectory mean obtained by robust input integration.

[0021] Figure 8 is the terminal distribution of the coordinates of the tilt-rotor aircraft under different disturbance levels in an embodiment of the present invention; Figure 8 (a) is the terminal distribution of the system XYZ coordinates obtained by integrating the deterministic input under the condition of 10% disturbance; Figure 8 (b) is the terminal distribution of the system XYZ coordinates obtained by robust input integration under 10% perturbation conditions; Figure 8 (c) is the terminal distribution of the system XYZ coordinates obtained by integrating the deterministic input under the condition of 20% disturbance; Figure 8 (d) is the terminal distribution of the system XYZ coordinates obtained by robust input integration under 20% perturbation conditions; Figure 8 (e) in the figure is the terminal distribution of the system XYZ coordinates obtained by integrating the deterministic input under the condition of 30% disturbance; Figure 8 (f) in the figure is the terminal distribution of the system's XYZ coordinates obtained by robust input integration under 30% perturbation conditions.

[0022] Figure 9 Comparison of the mean and standard deviation results of DPIM and MCS under 20% disturbance conditions in the embodiment of the present invention; Figure 9 (a) is the mean change curve of the system X coordinate calculated by the two methods under this condition; Figure 9 (b) is the standard deviation curve of the system X coordinate calculated by the two methods under this condition; Figure 9 (c) is the mean change curve of the system Y coordinate calculated by the two methods under this condition; Figure 9 (d) is the standard deviation curve of the system Y coordinate calculated by the two methods under this condition; Figure 9 (e) is the mean change curve of the system Z coordinate calculated by the two methods under this condition; Figure 9 (f) in the figure is the standard deviation curve of the system Z coordinate calculated by the two methods under this condition.

[0023] In the figure: 1 fuselage, 2 left nacelle, 3 right nacelle, 4 left rotor, 5 right rotor, 6 lower rotor, 7 upper rotor. DETAILED DESCRIPTION

[0024] The present invention will be further described below with reference to the accompanying drawings and specific embodiments. The enumerated embodiments are only used to explain the present invention and are not used to limit the scope of the present invention.

[0025] Considering the uncertainty in aerodynamic parameters during the transition phase of a tiltrotor aircraft in hover, the lift and reaction torque coefficient polynomials fitted by two-variable polynomials are shown in Tables 1 and 2, and the aerodynamic parameters that describe the linear relationship with angle of attack are shown in Table 3.

[0026] Table 1, lift coefficient

[0027]

[0028] Table 2, Anti-torque coefficient polynomials

[0029]

[0030] Table 3. Parameters of linear relationship of angle of attack

[0031]

[0032] This embodiment provides a tiltrotor aircraft robust trajectory optimization method based on a direct probability integration method, comprising the following steps:

[0033] Step 1: Consider the tiltrotor aircraft as a tiltrotor aircraft composed of seven parts and derive the multi-body dynamic model of the tiltrotor aircraft; specifically:

[0034] Step 1-1: Generalized coordinate description of motion;

[0035] The configuration of the tilt-rotor aircraft is as follows Figure 2 As shown, the coordinate diagram of the tilt-rotor aircraft is as follows Figure 3 As shown. The generalized coordinates that include all possible degrees of freedom are used to describe the motion state of the tilt-rotor aircraft, and the Euler angle is used as the generalized coordinates to describe the center of mass attitude of each component of the tilt-rotor aircraft. Therefore, The position coordinates of a rigid body in space are represented by vectors Its orientation relative to a fixed reference frame is expressed using the Cardan angle vector Indicates. Among them, 、 and Denote the pitch angle, roll angle and yaw angle respectively, with the subscripts Indicates the rigid body; the rotation order is: first around Axis rotation , then go around the new Axis rotation , and finally around the new Axis rotation . The generalized coordinates of each rigid body can be expressed as Represent the fuselage 1, left nacelle 2, right nacelle 3, left rotor 4, right rotor 5, lower rotor 6, upper rotor 7, the seven rigid body components of the integrated tilt-rotor aircraft, and the generalized coordinate vector of the tilt-rotor aircraft as a whole Expressed as:

[0036] (1)

[0037] In a time interval An infinitesimal rotation of a rigid body It can be decomposed into infinitesimal rotations around three axes, expressed as ,in 、 and They are axis, Axis and Unit direction vector on the axis; Indicates the A rigid body around Angle of shaft rotation; Indicates the A rigid body around Angle of shaft rotation; Indicates the A rigid body around The angle of the axis rotation; the rotation order is: first around Axis rotation , around the new Axis rotation , and finally around the new Axis rotation ; Indicates the The angle of rotation of the rigid body. Divide each term by And order Tends to 0, using the coefficient matrix to represent , we can get:

[0038] (2)

[0039] Among them, the instantaneous angular velocity Expressed in terms of the derivative of the Cardan angle; represents the angular velocity vector; Representing variables derivative with respect to time; is the variational symbol; the coefficient matrix and the angle vector The definition is as follows:

[0040] (3)

[0041] Step 1-2: Aerodynamic expression of rotor and wing;

[0042] Since the blade aspect ratio of the tilt-rotor aircraft is different from that of the conventional helicopter, the aerodynamic model can be simplified. The aerodynamic force acting on the tilt-rotor aircraft is as follows: Figure 4 In order to emphasize the aerodynamic characteristics of the rotor, the effects of the advance ratio and blade tip angle of attack on the thrust coefficient are mainly considered.

[0043] Currently, most tiltrotor aircraft use DC motors, so the thrust of the rotor varies with the rotor speed. The rotor aerodynamic model can be simplified using blade element theory as follows:

[0044] (4)

[0045] in, is the thrust generated by the rotor, is the tensile coefficient, is the rotor speed.

[0046] The bivariate polynomial fitting method is used to fit the changes of aerodynamic parameters with advance ratio and wingtip angle of attack. The fitting results are as follows:

[0047] (5)

[0048] in, is obtained by polynomial fitting and function; is the wingtip angle of attack; is the forward ratio; yes The power of yes The power of is the fitting coefficient; Wingtip angle of attack power; Forward ratio Power.

[0049] The aerodynamic forces considered for tiltrotor aircraft also include lift, drag, and pitching moment. The aerodynamic model of the wing and fuselage 1 is expressed using the following equation:

[0050] (6)

[0051] in, 、 and They represent the lift, drag and pitching moment acting on the tiltrotor aircraft respectively; is the wing angle of attack; is the forward true airspeed; is the air density; is the lift reference area; is the resistance reference area; is the reference chord length, 、 and They represent the lift coefficient at zero angle of attack, the drag coefficient at zero lift, and the pitching moment coefficient at zero angle of attack respectively; 、 and They represent the rates of change of lift coefficient, drag coefficient and pitching moment with angle of attack respectively.

[0052] The above aerodynamic forces are calculated in their respective airflow coordinate systems and act on the tiltrotor aircraft space coordinate system. The external force on a rigid body is expressed as:

[0053] (7)

[0054] in, It acts on The external force vector on a rigid body; It is The transformation matrix of the rigid body's center of mass coordinate system relative to the inertial system; It acts on The aerodynamic force vector on the center of mass coordinate system of a rigid body, including lift ,resistance and pitching moment The weight; It is the first The gravity vector of a rigid body.

[0055] Steps 1-3: Build a multi-body dynamics model;

[0056] First, a dynamic model is constructed for each component of the tiltrotor aircraft, such as the fuselage 1, the nacelle, and the rigid bodies in the rotor tiltrotor aircraft, such as Figure 5 As shown, the constraint force is then introduced using Lagrange multipliers, and the modular assembly method is adopted to finally obtain the complete multi-body dynamic model of the rotor tiltrotor aircraft.

[0057] The tilt-rotor aircraft can be regarded as a multi-body system composed of multiple rigid bodies. Therefore, the generalized dynamic equations are applicable to each component of the tilt-rotor aircraft. According to the principle of virtual work, the first The dynamic variational equation of a rigid body can be expressed as:

[0058] (8)

[0059] in, It is The inertia matrix of a rigid body, ,in, For the The mass of a rigid body, For the The moment of inertia of a rigid body about the x-axis, For the The moment of inertia of a rigid body about the y-axis, For the The moment of inertia of a rigid body around the z-axis; It is The inertia matrix of a rigid body, ,in, is a diagonal matrix; It is the effect on the tilt-rotor aircraft The external force vector on a rigid body; For the The angular acceleration vector of a rigid body; For the The center-of-mass acceleration vector of a rigid body; For the The transpose of the variation of the angular acceleration of a rigid body; For the The transpose of the variation of the velocity of the center of mass of a rigid body.

[0060] Substituting the coordinates in formula (1) into formula (8) yields:

[0061] (9)

[0062] in, Indicates the The generalized velocity of a rigid body; Indicates the The coordinate transformation matrix of a rigid body; Indicates the The generalized acceleration of a rigid body; Indicates the Generalized velocity-dependent terms for rigid bodies; Indicates the The inertial force vector of a rigid body due to rotation; Indicates the effect on The external force vector on a rigid body; the matrix , and Expressed as:

[0063] (10)

[0064] (11)

[0065] (12)

[0066] in, Indicates the The roll angle of a rigid body; Indicates the The yaw angle of a rigid body; Indicates The first-order derivative of the coefficient matrix related to the rigid body attitude angle with respect to time; Indicates the The angular velocity vector of a rigid body.

[0067] After simplifying the above formula (9), the virtual power equation satisfied by the generalized coordinates of the tilt-rotor aircraft is obtained, as shown in formula (13):

[0068] (13)

[0069] Among them, the matrix and Expressed as .matrix is the mass matrix of the tiltrotor aircraft, is the generalized external force vector of the tiltrotor aircraft, is the generalized acceleration vector of the tiltrotor aircraft.

[0070] In tiltrotor aircraft, represents the inertial coordinate system, They represent the coordinate systems fixed on the mass centers of the fuselage 1, the left nacelle 2, the right nacelle 3, the left rotor 4, the right rotor 5, the lower rotor 6, and the upper rotor 7, respectively. Represents their position vectors in the inertial system. The fuselage 1 and the nacelle are connected by a rotating hinge. The connection point coincides with the center of mass of the nacelle. The rotation axis of the nacelle coincides with the center of mass of the nacelle coordinate system. The axis coincides with the rotor, the nacelle and the rotor are also connected by a rotating hinge, the connection point coincides with the center of mass of the rotor, and the rotation axis of the rotor coincides with the center of mass of the rotor coordinate system. Since the tilt-rotor aircraft is bilaterally symmetrical, taking the hinge connection between the fuselage 1 and the left nacelle 2 as an example, assuming that the connection point between the fuselage 1 and the left nacelle 2 is point P, is the position vector of point P in the coordinate system of fuselage 1. According to the characteristics of the rotary hinge, during the motion process, the relative position of point P in the coordinate system of fuselage 1 and the coordinate system of left nacelle 2 remains unchanged, and the rotation axis of left nacelle 2 The axis is always perpendicular to the coordinate system of the fuselage 1 Plane. Specifically in this embodiment, for Figure 4 The tilt mechanism of the tilt-rotor aircraft is shown in the figure. represents the inertial coordinate system, They represent the coordinate systems defined at the center of mass of the fuselage 1, the left nacelle 2, and the left rotor 4, respectively. Represents their position vectors in the inertial system. The fuselage 1 and the left nacelle 2 are connected by a rotating hinge at P. Point P coincides with the center of mass of the left nacelle 2. The rotation axis of the left nacelle 2 coincides with the coordinate system of the left nacelle 2. The left nacelle 2 is connected to the left rotor 4 at U through a rotating hinge. Point U coincides with the center of mass of the left rotor 4. The rotation axis of the left rotor 4 coincides with the coordinate system of the left rotor 4. Axes coincide. is the position vector of point P in the coordinate system of fuselage 1, is the position vector of point U in the left nacelle 2 coordinate system. According to the characteristics of the rotary hinge of the left nacelle 2 of the fuselage 1, during the movement, the relative position of point P in the coordinate system of the fuselage 1 and the coordinate system of the left nacelle 2 remains unchanged, and the rotation axis of the left nacelle 2 The axis is always perpendicular to the coordinate system of the fuselage 1 flat.

[0071] Therefore, the attitude constraint equation between the fuselage 1 and the left nacelle 2 is Expressed as:

[0072] (14)

[0073] in, and are the transformation matrices of fuselage 1 and left nacelle 2 relative to the inertial system; 、 and They are corresponding 、 and The unit direction vector of the coordinate axis; and are the pitch angles of fuselage 1 and nacelle, respectively; is the angle between the fuselage 1 and the nacelle that changes with time; is the position vector of the fuselage 1 in the inertial coordinate system; is the position vector of the left nacelle 2 in the inertial coordinate system; It is the position vector of point P in the coordinate system of fuselage 1.

[0074] Similarly, the posture constraint equations of the hinge connections between other adjacent parts can be derived as follows:

[0075] (15)

[0076] (16)

[0077] (17)

[0078] (18)

[0079] (19)

[0080] in, represents the attitude constraint equation between the fuselage 1 and the right nacelle 3; and They represent the attitude constraint equations between the left nacelle 2 and the left rotor 4, and between the right nacelle 3 and the right rotor 5 respectively; and Respectively represent the attitude constraint equations between the fuselage 1 and the lower rotor 6 behind the fuselage 1, and between the lower rotor 6 and the upper rotor 7; vectors with subscripts Represents the unit direction vector corresponding to the coordinate axis; 、 and are the yaw angles of the fuselage 1, nacelle, and rotor center of mass, respectively; represents the change of rotor speed over time; Indicates the The transformation matrix of the local coordinate system of a rigid body relative to the inertial system is defined as:

[0081] (20)

[0082] After combining the attitude constraint equations of all the rotary hinges in the tilt-rotor aircraft, the complete constraint equation of the tilt-rotor aircraft can be expressed as:

[0083] (twenty one)

[0084] The generalized coordinates of the tiltrotor aircraft subject to the above constraint equations are It is not independent, the virtual velocity of the tiltrotor aircraft satisfies ,in is the Jacobian matrix of the constraint equations for the tiltrotor aircraft, which means that any vector orthogonal to the imaginary velocity must be The linear combination of the columns of and the virtual velocity of the tiltrotor aircraft Orthogonal, so we can get:

[0085] (twenty two)

[0086] in, is a Lagrange multiplier vector whose dimension is the same as the number of constraint equations; is the Jacobian matrix of the tiltrotor aircraft constraint equations; is the generalized acceleration vector of the tiltrotor aircraft.

[0087] Combining formula (21) and formula (22), the dynamic model of the tilt-rotor aircraft can be expressed as:

[0088] (twenty three)

[0089] in, is the complete constraint equation for the tiltrotor aircraft.

[0090] Step 2: Formulate a deterministic trajectory optimization problem based on the tiltrotor aircraft's dynamic equations, mechanical constraints, and mission requirements.

[0091] Step 2-1: Define the design variables of the optimization problem;

[0092] Tilt-rotor aircraft mainly achieve transitions between flight modes by tilting the rotor through the nacelle. During the transition process, the nacelle gradually tilts from a perpendicular position relative to the fuselage 1 to a position parallel to the fuselage 1, completing the switch from helicopter mode to fixed-wing mode. In this process, the lift of the tilt-rotor aircraft gradually changes from being provided by the rotor to being provided by the wing, which will cause significant fluctuations in the flight state, affecting flight performance and safety. Therefore, the transition mode is a key stage in the transition between helicopters and fixed-wing aircraft, and is of great significance to the performance optimization of tilt-rotor aircraft.

[0093] The optimization of the tilt drive law of the nacelle of a tilt-rotor aircraft is different from the optimization of the structural parameters of the tilt-rotor aircraft in multi-body dynamics. Since the continuous drive law curve needs to be discretized, it is crucial to select appropriate design variables. Considering that the B-spline curve has the characteristics of local adjustability, recursion and differentiability, the B-spline curve is used to approximate the tilt change of the nacelle and the change of the rotor speed. In formula (14), the tilt law of the nacelle is defined as , which is described by the uniform cubic B-spline function, and the control points are used as the design variables of the optimization problem. The expression of the B-spline curve is shown in formula (24):

[0094] (twenty four)

[0095] in, Represents the B-spline curve at time The function value at ; represents a discrete time point; Represents the control point vector of the B-spline curve; Represents the index parameter, which is used to distinguish two different B-spline curves; The index representing the time segment.

[0096] According to formula (24), when hour, ,in represents the function of the nacelle tilt angle changing with time, .when hour, ,in A function representing the rotor speed changing with time Considering the maneuverability of the tilt-rotor aircraft, the tilt transition time is set to 3 seconds, that is, .

[0097] Therefore, the design variables of the tiltrotor aircraft are , where the first six variables represent the control points of the cabin tilt change, and the last six variables represent the control points of the rotor speed. The initial values ​​of the design variables are recorded as , the upper bound of the design variable is denoted as , the lower bound is denoted as , respectively:

[0098] (25)

[0099] In time interval The nacelle tilt is a function of time and the rotor speed as a function of time In the constraint equation is converted into B-spline control points By applying the piecewise drive law, the motion of the tiltrotor aircraft is described by a set of differential algebraic equations. The dynamic response of the tiltrotor aircraft is also affected by the design variables Therefore, the reformulated dynamic equation can be expressed in a constrained form as:

[0100] (26)

[0101] Step 2-2: Determine the objective function and construct the optimization problem;

[0102] The objective function of the optimization problem is defined as follows:

[0103] (27)

[0104] in, is the objective function; and are the initial and final moments of the tiltrotor aircraft’s transition phase; and Indicates the instantaneous flight altitude and initial altitude of the aircraft; The center of mass is Speed ​​in direction; and are the pitch angles of fuselage 1 and nacelle respectively.

[0105] The goal of the optimization problem is to minimize the change in the flight altitude of the tilt-rotor aircraft during the transition phase. The speed in the direction is minimum, and finally the angle between the fuselage 1 and the nacelle becomes 90 degrees.

[0106] By combining the objective function and the dynamic constraints, the optimization problem of the tiltrotor aircraft model is defined as follows:

[0107] (28)

[0108] in, is the objective function; is the design variable; and They are design variables The upper and lower bounds of . The dynamic equations of a tiltrotor aircraft are described by differential algebraic equations.

[0109] Step 3: Introduce uncertainty variables into the deterministic trajectory optimization problem to obtain the uncertain trajectory optimization problem, and transform it into a robust trajectory optimization problem;

[0110] The uncertainties faced by the tiltrotor aircraft during the transition phase mainly include the uncertainty of dynamic parameters such as atmospheric density and aerodynamic parameters. Assuming that there is uncertainty in the lift-related parameters of the aerodynamic force acting on the tiltrotor aircraft, it can be specifically expressed as:

[0111] (29)

[0112] Among them, the lift coefficient Then a single random variable is added ,Bundle is represented as a random variable uniformly distributed in a certain interval, such as ; Indicates lift; Indicates the air density; Indicates forward true airspeed; represents the lift reference area; represents the lift coefficient; represents the wing angle of attack.

[0113] When random variables are added to the lift calculation equation After that, the original deterministic optimization problem shown in formula (28) is transformed into an optimization problem with uncertainty, as shown in formula (30):

[0114] (30)

[0115] Compared with the deterministic optimization problem shown in formula (28), a variable used to describe uncertainty is introduced in the uncertainty optimization problem shown in formula (30): Taking into account the influence of probabilistic uncertainty, the objective function (as shown in Equation (27)), the dynamic equations (as shown in Equation (23)), and the constraints (as shown in Equation (21)) are not fixed but have random properties. This is because the tiltrotor aircraft state and control inputs may be affected by uncertain factors. In the process of uncertainty propagation, they can be regarded as functions of random variables.

[0116] Because of the uncertain variables , the uncertainty optimization problem shown in formula (30) cannot be directly solved numerically. In order to fully consider the impact of uncertainty on the solution of the optimization problem, the following robust optimization formula is proposed:

[0117] (31)

[0118] in, and denote the mean and standard deviation of the random variable respectively; Indicates the threshold that the standard deviation of the constraint function should meet.

[0119] As shown in formula (31), the threshold value of the standard deviation of the constraint function is satisfied Set to zero vector, this is because when using the direct probability integration method to solve the trajectory optimization problem with uncertainty, it is actually solving The sub-deterministic trajectory optimization problem is similar to the Monte Carlo method. Therefore, the goal is to ensure that each solution satisfies the constraints under a given level of uncertainty and robustness. For example, the solution to the problem shown in Equation (31) must strictly obey the random constraint function and minimize the expectation of the objective function.

[0120] Step 4: Use direct probability integration method to calculate the uncertainty variables. Subsampling,quantizes the random function in the robust trajectory optimization problem;

[0121] Step 4-1: Sampling by direct probability integration method Representative points;

[0122] First, for a group dimensional continuous random variable, the possible random events in the sample space of this random variable may be infinite. According to the basic idea of ​​Lebesgue integral, infinite random events must be approximated by finite random events. In a finite number of dimensional Euclidean spaces, these finite random events are called representative points. The corresponding probability measures are calculated from the representative regions of these points, and the sum of the probabilities of each representative point is 1. Generating representative points and assigning probabilities can be achieved using a point selection method based on generalized F-deviation (GF-deviation). When selecting representative points, it is necessary to minimize the GF-deviation. This is achieved through a point set rearrangement method: first, an initial point set is generated, and then the initial point set is rearranged to reduce the GF-deviation. Based on experience, the Sobol sequence is often used as the initial point set.

[0123] In order to reduce the GF-bias, the following transformation is required. In the first step, the initial point set It is obtained from the Sobol set of the unit hypercube by formula (32) Obtained in:

[0124] (32)

[0125] in, is the joint distribution function The inverse function of dimension; Indicates the The sample point in The value of the dimension; Indicates the first The point in The value of the dimension.

[0126] Then, make the following changes in each dimension of the random variable so that The probability of assigning values ​​to each point is close, as shown in formula (33):

[0127] (33)

[0128] in, is an indicator function that satisfies ; After transformation The sample point in The value of the dimension; For the The sample point in The value of the dimension.

[0129] Finally, through Calculate The probability of assigning a representative point Then perform the following transformations:

[0130] (34)

[0131] After the above transformation, As the representative point set with the smallest GF-deviation finally adopted.

[0132] In addition, the probability space of the input random variables is divided into a set of non-overlapping representative regions using a partitioning technique based on Voronoi cells. ,Right now:

[0133] (35)

[0134] in, Indicates the The representative point of a Voronoi cell; Indicates the The representative point of a Voronoi cell; Indicates the Voronoi cells; Indicates the Voronoi cells; express Euclidean space.

[0135] At the representative point, the probability of the output random vector is equal to the probability carried by the point, which reflects the principle of conservation of probability, and the PDF of the output random vector is all possible responses of the input random vector, that is, the sum of the PDFs of all discretized random events.

[0136] Step 4-2: Quantize the random function;

[0137] According to the robust trajectory optimization problem shown in formula (31) obtained in step 3, when solving it, it is necessary to convert the objective function and constraint function containing random variables into deterministic statistics, such as mean and variance. This process relies on the quantization mechanism of the direct probability integration method.

[0138] Assuming that both the objective and constraint functions are ordinary nonlinear functions with finite variance, their statistical properties can be captured by weighted summation of representative points. The mean of the objective and constraint functions can be expressed as the sum of the product of the function values ​​at each representative point and the corresponding probability, while the standard deviation is calculated as the weighted sum of the squares of the mean and the deviations from the function values ​​at each point. Uncertain variables in the dynamic equations do not require additional transformation; simply substitute each representative point in turn during the solution to obtain the corresponding deterministic solution.

[0139] Assume that the direct probability integration method is used to select Representative points, each representative point The corresponding distribution probability is , the mean and standard deviation of the objective function and constraint function can be expressed as:

[0140] (36)

[0141] (37)

[0142] in, represents the mean of the objective function; Indicates that the Representative points Substitute the obtained value of the objective function; Indicates the Representative points The corresponding distribution probability; represents the mean of the constraint function; represents the standard deviation of the constraint function; Indicates that the Representative points Substitute the obtained value of the constraint function.

[0143] Step 5: Substitute the transformed random function into the robust trajectory optimization problem to obtain A deterministic trajectory optimization problem is solved to obtain the statistical characteristics of the tiltrotor aircraft state and control input;

[0144] Substituting the mean of the objective function shown in formula (36) and the mean and standard deviation of the constraint function shown in formula (37) into formula (31), the robust trajectory optimization problem shown in formula (31) is transformed into a deterministic trajectory optimization problem that can be directly solved, as shown in formula (38).

[0145] (38)

[0146] At this point, the robust trajectory optimization problem of the tiltrotor aircraft has been transformed into a deterministic optimization problem. Finally, the deterministic trajectory optimization problem shown in formula (38) is solved to obtain the optimal control input and statistical characteristics of the state quantity of the tiltrotor aircraft.

[0147] In this embodiment, the robust trajectory optimization problem containing random functions is directly converted into A deterministic trajectory optimization problem, compared with the polynomial chaos expansion method, does not require dimensionality increase;

[0148] In this embodiment, the direct probability integration method differs from the random sampling of the traditional Monte Carlo method. The direct probability integration method introduces a point selection technique based on GF-deviation and a Dirac function smoothing technique. Based on the principle of probability conservation, the method adopts a numerical integration method during integration. The method reflects the phenomenon of unequal weights of integration points by dividing the probability space, thereby greatly reducing the number of sample points used.

[0149] In this embodiment, the tilt-rotor aircraft satisfies the following assumptions: 1) the earth is regarded as an inertial system; 2) the aircraft is regarded as a rigid body without elastic deformation; 3) the aircraft is bilaterally symmetrical, and the center of mass of each rigid body remains unchanged; 4) the aerodynamic coupling effect between the rotor and the wing is ignored.

[0150] The following example uses the robust trajectory optimization problem of a tiltrotor aircraft completing a transition mode in a hovering state with aerodynamic parameter uncertainty to verify the optimization algorithm provided in this example.

[0151] The parameters of the seven components of the tilt-rotor aircraft model in this embodiment are shown in Table 4.

[0152] Table 4. Initial position, attitude, and physical parameters of the tiltrotor aircraft.

[0153]

[0154] This embodiment discusses the optimization results when the lift coefficient is subject to different degrees of disturbance during the transition of a tiltrotor aircraft from helicopter mode to fixed-wing mode in a hovering state. The uncertainty levels are set to a uniform distribution of 10%, 20%, and 30%, respectively, with other conditions remaining the same.

[0155] The control input curves obtained under different disturbance conditions in this embodiment are shown in the figure, where Figure 6 (a) in the figure represents the control curve of the inclination angle. Figure 6 (b) in the figure shows the speed control curve. As can be seen from the figure, there is a clear difference between the input curve in the deterministic case and the input curve with uncertainty, and as the degree of uncertainty increases, the difference between the input curves gradually decreases.

[0156] The lift coefficient is randomly sampled at 1000 sample points by Monte Carlo simulation. Figure 6 By integrating the four different input curves obtained from (a) and (b) on the dynamic equation of the tilt-rotor aircraft, the dynamic response of the tilt-rotor aircraft can be obtained. The mean value of the final 1000 flight trajectories changes as shown in the figure. Figure 7 As shown, Figure 7 (a) in the figure represents the trajectory mean obtained by integrating the deterministic input when the uncertainty of the aerodynamic parameters is 10%, 20%, and 30%, respectively. Figure 7 (b) in the figure represents the mean value of the robust solution obtained by integrating the three cases. Figure 7 As shown in Figures (a) and (b), the flight trajectory is increasingly affected as the uncertainty of the tiltrotor's lift coefficient increases. In the three mean trajectories obtained through robust integration, the Z coordinate of the tiltrotor's final state gradually increases from approximately 0.1m to approximately 0.4m. However, the trajectory obtained through deterministic input integration shows no regular trend. When the uncertainty is 10%, the final altitude is approximately 0.02m, when the uncertainty is 20%, the final altitude is approximately 0.06m, and when the uncertainty increases to 30%, the final altitude drops to approximately -0.02m.

[0157] Figure 8 Figures (a), (b), (c), (d), (e), and (f) show the possible landing points of the three coordinates of the tiltrotor aircraft fuselage 1 obtained from 1000 Monte Carlo simulations in this embodiment. (a), (c), and (e) represent the distributions of the final state obtained by integrating the deterministic input with uncertainty levels of 10%, 20%, and 30%, respectively, while (b), (d), and (f) represent the distributions of the robust solution obtained by integrating the solution under the three perturbations. The more dispersed the final landing points, the greater the volatility of the final state of the tiltrotor aircraft and the worse its stability; conversely, the better its stability. The means and standard deviations of these distributions are listed in Table 5.

[0158] Table 5. Statistical characteristics of the distribution of the three coordinate terminals under three disturbance conditions

[0159]

[0160] Figure 8 (a), (b), (c), (d), (e), (f) and This paper demonstrates a clear difference in the distribution of the terminal results obtained by integrating the robust input with uncertainty and the deterministic input without uncertainty. The results show that the terminal state obtained by integrating the deterministic input fluctuates much more than the robust result, demonstrating the effectiveness and accuracy of the proposed method for solving robust trajectory optimization problems. The X-coordinate represents the lateral offset of the tiltrotor during the tilting process. Under the three uncertainty conditions, the mean and standard deviation of the offset of the deterministic result are both greater than those of the robust result. The Y-coordinate represents the forward flight distance of the tiltrotor. The forward flight distance of the deterministic result is slightly greater than that of the robust result, but its standard deviation increases with increasing uncertainty. Under 10% perturbation, the standard deviation of the deterministic result is approximately 2.9 times that of the robust result, increasing to 3.9 and 5.5 times for 20% and 30% perturbations, respectively. Similarly, the standard deviation of the Z-coordinate of the deterministic result increases by 60%, 89%, and 125% compared to the robust result under the three conditions. From this, we can conclude that as the degree of uncertainty increases, the adaptability of the deterministic results becomes worse, while the robust results, because they take the impact of uncertainty into account, are more adaptable to disturbances, and their advantages over the deterministic results are becoming more and more obvious.

[0161] Taking the case of 20% disturbance in the hovering state as an example, the direct probability integral method (DPIM) proposed in this invention is compared with the Monte Carlo simulation (MCS). By comparing the means and standard deviations of the three coordinates obtained by the two methods, it is found that Figure 9 The results show that the mean and standard deviation of the results obtained by this method are very consistent with MCS throughout the entire time period, indicating that the solution accuracy of DPIM is very close to that of MCS. However, DPIM requires fewer sample points than MCS, greatly improving the solution efficiency, highlighting the advantages of DPIM in solving robust trajectory optimization problems.

[0162] The above-described embodiments merely express the implementation methods of the present invention, but should not be understood as limiting the scope of the patent of the present invention. It should be pointed out that for those skilled in the art, several variations and improvements can be made without departing from the concept of the present invention, and these all fall within the scope of protection of the present invention.

Claims

1. A robust trajectory optimization method for a tiltrotor aircraft based on a direct probability integration method, characterized in that: The optimization method comprises the following steps: Step 1: Consider the tiltrotor aircraft as a tiltrotor aircraft composed of seven parts, and derive a multi-body dynamic model of the tiltrotor aircraft; Step 2: Construct a deterministic trajectory optimization problem based on the dynamic equations, mechanical constraints, and mission requirements of the tiltrotor aircraft. Step 3: Introduce uncertainty variables into the deterministic trajectory optimization problem to obtain the uncertain trajectory optimization problem, and transform it into a robust trajectory optimization problem; Step 4: Use direct probability integration method to calculate the uncertainty variables. Subsampling,quantizes the random function in the robust trajectory optimization problem; Step 5: Substitute the transformed random function into the robust trajectory optimization problem to obtain A deterministic trajectory optimization problem is solved to obtain the statistical characteristics of the tiltrotor aircraft state and control input.

2. The method for robust trajectory optimization of a tiltrotor aircraft based on direct probability integration method according to claim 1, characterized in that: The step 1 is specifically as follows: Step 1-1: Generalized coordinate description of motion: The generalized coordinates are used to describe the motion state of the tilt-rotor aircraft, and the Euler angle is used as the generalized coordinates to describe the center of mass attitude of the tilt-rotor aircraft components; then The position coordinates of a rigid body in space are represented by vectors Its orientation relative to a fixed reference frame is expressed using the Cardan angle vector Indicates; among them, , and Denote the pitch angle, roll angle and yaw angle respectively, with the subscripts Indicates the The rotation order is: first around Axis rotation , around the new Axis rotation , and finally around the new Axis rotation ; Express the generalized coordinates of each rigid body as Represent the fuselage, left nacelle, right nacelle, left rotor, right rotor, lower rotor, upper rotor, seven rigid body parts of the integrated tilt-rotor aircraft, and the generalized coordinate vector of the tilt-rotor aircraft as a whole Expressed as: (1) ； In a time interval An infinitesimal rotation of a rigid body Decomposed into infinitesimal rotations around three axes, expressed as ,in 、 and They are axis, Axis and Unit direction vector on the axis; Indicates the A rigid body around Angle of shaft rotation; Indicates the A rigid body around Angle of shaft rotation; Indicates the A rigid body around The angle of the axis rotation; the rotation order is: first around Axis rotation , around the new Axis rotation , and finally around the new Axis rotation ; Indicates the The angle of rotation of the rigid body; divide each term by And order Tends to 0, using the coefficient matrix to represent ,get: (2) ； in, represents the angular velocity vector; is a variable The derivative with respect to time represents the instantaneous angular velocity; is the variational symbol; the coefficient matrix and the angle vector The definition is as follows: (3) ； Step 1-2: Aerodynamic expression of rotor and wing: The rotor aerodynamic model is simplified using blade element theory as follows: (4) ； in, is the thrust generated by the rotor, is the tensile coefficient, is the rotor speed; The bivariate polynomial fitting method is used to fit the changes of aerodynamic parameters with advance ratio and wingtip angle of attack. The fitting results are as follows: (5) ； in, is obtained by polynomial fitting and function; is the wingtip angle of attack; is the forward ratio; yes The power of yes The power of is the fitting coefficient; Wingtip angle of attack power; Forward ratio power; The aerodynamic forces considered for tiltrotor aircraft also include lift, drag, and pitching moment; the aerodynamic model of the wing and fuselage is expressed by the following equations: (6) ； in, 、 and They represent the lift, drag and pitching moment acting on the tiltrotor aircraft respectively; is the wing angle of attack; is the forward true airspeed; is the air density; is the lift reference area; is the resistance reference area; is the reference chord length, 、 and They represent the lift coefficient at zero angle of attack, the drag coefficient at zero lift, and the pitching moment coefficient at zero angle of attack respectively; 、 and They represent the rates of change of lift coefficient, drag coefficient and pitching moment with angle of attack respectively; The above aerodynamic forces are calculated in their respective airflow coordinate systems and act on the tiltrotor aircraft space coordinate system. The external force on a rigid body is expressed as: (7) ； in, It acts on The external force vector on a rigid body; It is The transformation matrix of the rigid body's center of mass coordinate system relative to the inertial system; It acts on The aerodynamic force vector on the center of mass coordinate system of a rigid body, including lift ,resistance and pitching moment The weight; It is the first The gravity vector of a rigid body; Steps 1-3: Build a multi-body dynamics model; Firstly, a dynamic model is constructed for each rigid component of the tiltrotor aircraft. Then, the constraint force is introduced using Lagrange multipliers and a modular assembly method is adopted to finally obtain a complete multi-body dynamic model of the tiltrotor aircraft.

3. The method for robust trajectory optimization of a tiltrotor aircraft based on direct probability integration method according to claim 2, characterized in that: The steps 1-3 are as follows: Tilt-rotor aircraft The dynamic variational equation of a rigid body is expressed as: (8) ； in, It is The inertia matrix of a rigid body, ,in, For the The mass of a rigid body, For the The moment of inertia of a rigid body about the x-axis, For the The moment of inertia of a rigid body about the y-axis, For the The moment of inertia of a rigid body around the z-axis; It is The inertia matrix of a rigid body, ,in, is a diagonal matrix; It is the effect on the tilt-rotor aircraft The external force vector on a rigid body; For the The angular acceleration vector of a rigid body; For the The center-of-mass acceleration vector of a rigid body; For the The transpose of the variation of the angular acceleration of a rigid body; For the The transpose of the variation of the velocity of the center of mass of a rigid body; Substituting the coordinates in formula (1) into formula (8) yields: (9) ； in, Indicates the The generalized velocity of a rigid body; Indicates the The coordinate transformation matrix of a rigid body; Indicates the The generalized acceleration of a rigid body; Indicates the Generalized velocity-dependent terms for rigid bodies; Indicates the The inertial force vector of a rigid body due to rotation; Indicates the effect on The external force vector on a rigid body; the matrix , and Expressed as: (10) ； (11) ； (12) ； in, Indicates the The roll angle of a rigid body; Indicates the The yaw angle of a rigid body; Indicates the The first-order derivative of the coefficient matrix related to the rigid body attitude angle with respect to time; Indicates the The angular velocity vector of a rigid body; After simplifying the above formula (9), the virtual power equation satisfied by the generalized coordinates of the tilt-rotor aircraft is obtained, as shown in formula (13): (13) ； Among them, the matrix and Expressed as ,matrix is the mass matrix of the tiltrotor aircraft, is the generalized external force vector of the tiltrotor aircraft, is the generalized acceleration vector of the tiltrotor aircraft; Attitude constraint equation between fuselage and left nacelle Expressed as: (14) ； in, and are the transformation matrices of the fuselage and the left nacelle relative to the inertial system; 、 and They are corresponding 、 and The unit direction vector of the coordinate axis; and are the pitch angles of the fuselage and nacelle, respectively; is the time-varying angle between the fuselage and the nacelle; is the position vector of the fuselage in the inertial coordinate system; is the position vector of the left nacelle in the inertial coordinate system; is the position vector of point P in the fuselage coordinate system; Similarly, the posture constraint equations of the hinge connections between other adjacent rigid components are derived as follows: (15) ； (16) ； (17) ； (18) ； (19) ； in, represents the attitude constraint equation between the fuselage and the right nacelle; and They represent the attitude constraint equations between the left nacelle and the left rotor, and between the right nacelle and the right rotor respectively; and Respectively represent the attitude constraint equations between the fuselage and the lower rotor behind the fuselage, and between the lower rotor and the upper rotor; vectors with subscripts Represents the unit direction vector corresponding to the coordinate axis; 、 and are the yaw angles of the fuselage, nacelle, and rotor center of mass, respectively; represents the change of rotor speed over time; Indicates the The transformation matrix of the local coordinate system of a rigid body relative to the inertial system is defined as: (20) ； After combining the attitude constraint equations of all the rotary hinges in the tilt-rotor aircraft, the complete constraint equation of the tilt-rotor aircraft can be expressed as: (21) ； The virtual velocity of the tiltrotor aircraft satisfies ,in is the Jacobian matrix of the constraint equations for the tiltrotor aircraft, which means that any vector orthogonal to the imaginary velocity must be The linear combination of the columns of and the virtual velocity of the tiltrotor aircraft Orthogonal, so we get: (22) ； in, is a Lagrange multiplier vector whose dimension is the same as the number of constraint equations; is the Jacobian matrix of the tiltrotor aircraft constraint equations; is the generalized acceleration vector of the tiltrotor aircraft; Combining formula (21) and formula (22), the dynamic model of the tilt-rotor aircraft is expressed as: (23) ； in, is the complete constraint equation for the tiltrotor aircraft.

4. The method for robust trajectory optimization of a tiltrotor aircraft based on direct probability integration method according to claim 3, characterized in that: In the steps 1-3, in the tilt-rotor aircraft, represents the inertial coordinate system, Respectively represent the coordinate systems fixed on the center of mass of the fuselage, left nacelle, right nacelle, left rotor, right rotor, lower rotor and upper rotor, Denotes their position vectors in the inertial system, where The fuselage and the nacelle are connected by a rotating hinge, the connection point coincides with the center of mass of the nacelle, and the rotation axis of the nacelle is aligned with the center of mass of the nacelle coordinate system. The axis coincides with the rotor, the nacelle and the rotor are also connected by a rotating hinge, the connection point coincides with the center of mass of the rotor, and the rotation axis of the rotor coincides with the center of mass of the rotor coordinate system. The axes coincide; the tilt-rotor aircraft is bilaterally symmetrical, and is described by the hinge connection between the fuselage and the left nacelle. Assume that the connection point between the fuselage and the left nacelle is point P. is the position vector of point P in the fuselage coordinate system; according to the characteristics of the rotary hinge, during the movement, the relative position of point P in the fuselage coordinate system and the left nacelle coordinate system remains unchanged, and the rotation axis of the left nacelle is The axis is always perpendicular to the fuselage coordinate system flat.

5. The method for robust trajectory optimization of a tiltrotor aircraft based on direct probability integration method according to claim 3, characterized in that: The step 2 is specifically as follows: Step 2-1: Define the design variables of the optimization problem; The B-spline curve is used to approximate the nacelle tilt change and the rotor speed change; in formula (14), the nacelle tilt law is defined as , which is described by a uniform cubic B-spline function, with the control points as the design variables of the optimization problem. The expression of the B-spline curve is shown in formula (24): (24) ； in, Represents the B-spline curve at time The function value at ; represents a discrete time point; Represents the control point vector of the B-spline curve; Represents the index parameter, which is used to distinguish two different B-spline curves; The index representing the time segment; The design variables of the tiltrotor aircraft are , where the first six variables represent the control points of the cabin tilt change, and the last six variables represent the control points of the rotor speed; the initial values ​​of the design variables are recorded as , the upper bound of the design variable is denoted as , the lower bound is ; In time interval The nacelle tilt is a function of time and the rotor speed as a function of time In the constraint equation is converted into B-spline control points By applying the piecewise drive law, the motion of the tiltrotor aircraft is described by a set of differential algebraic equations; the dynamic response of the tiltrotor aircraft is also affected by the design variables The reformulated dynamic equation is expressed in constraint form as follows: (25) ； Step 2-2: Determine the objective function and construct the optimization problem; The objective function of the optimization problem is defined as follows: (26) ； in, is the objective function; and are the initial and final moments of the tiltrotor aircraft’s transition phase; and Indicates the instantaneous flight altitude and initial altitude of the aircraft; The center of mass is Speed ​​in direction; and are the pitch angles of the fuselage and nacelle, respectively; The goal of the optimization problem is to minimize the change in the flight altitude of the tilt-rotor aircraft during the transition phase. The speed in the direction is minimum and finally the angle between the fuselage and the nacelle becomes 90 degrees; By combining the objective function and the dynamic constraints, the optimization problem of the tiltrotor aircraft model is defined as follows: (27) ； in, is the objective function; is the design variable; and They are design variables The upper and lower bounds of The dynamic equations of a tiltrotor aircraft are described by differential algebraic equations.

6. The method for robust trajectory optimization of a tiltrotor aircraft based on direct probability integration method according to claim 5, characterized in that: In step 2-1, according to formula (24), when hour, ,in represents the function of the nacelle tilt angle changing with time, ;when hour, ,in A function representing the rotor speed changing with time ; Considering the maneuverability of the tilt-rotor aircraft, the tilt transition time is set to 3 seconds, that is, .

7. The method for robust trajectory optimization of a tiltrotor aircraft based on direct probability integration method according to claim 6, characterized in that: The step 3 is specifically as follows: Assume that there is uncertainty in the lift-related parameters of the aerodynamic force acting on the tilt-rotor aircraft, which can be expressed as follows: (28) ； Among them, the lift coefficient Then a single random variable is added ,Bundle Represented as a random variable uniformly distributed in a certain interval; Indicates lift; Indicates the air density; Indicates forward true airspeed; represents the lift reference area; represents the lift coefficient; represents the wing angle of attack; When random variables are added to the lift calculation equation After that, the deterministic optimization problem shown in formula (27) is transformed into an optimization problem with uncertainty, as shown in formula (29): (29) ； Taking into account the influence of probabilistic uncertainty, the objective function shown in formula (26), the dynamic equation shown in formula (23), and the constraints shown in formula (21) have random properties.

8. The method for robust trajectory optimization of a tiltrotor aircraft based on direct probability integration method according to claim 7, characterized in that: In step 3, in order to fully consider the impact of uncertainty on solving the optimization problem, the following robust optimization formula is proposed: (30) ； in, and denote the mean and standard deviation of the random variable respectively; Indicates the threshold that the standard deviation of the constraint function should meet; As shown in formula (30), the threshold value of the standard deviation of the constraint function is satisfied is set to the zero vector; the goal is to ensure that each solution satisfies the constraints under a given level of uncertainty and robustness, such as the solution to the problem shown in formula (30) obeys the random constraint function and minimizes the expectation of the objective function.

9. The method for robust trajectory optimization of a tiltrotor aircraft based on direct probability integration method according to claim 8, characterized in that: The step 4 is specifically as follows: Step 4-1: Sampling by direct probability integration method Representative points; Define a set dimensional continuous random variable, in In a finite-dimensional Euclidean space, finite random events are called representative points. The corresponding probability measure is calculated from the representative regions of the representative points, and the sum of the probabilities of each representative point is 1. The generation of representative points and the assignment of probabilities are achieved through a point selection method based on generalized F-deviation. When selecting representative points, the GF-deviation is minimized through a point set rearrangement method: first, an initial point set is generated, and then the initial point set is rearranged to reduce the GF-deviation. The Sobol sequence is used as the initial point set. First, to reduce the GF-deviation, the following transformation is used; the initial point set It is obtained from the Sobol set of the unit hypercube by formula (31) Obtained in: (31) ； in, is the joint distribution function The inverse function of dimension; Indicates the The sample point in The value of the dimension; Indicates the first The point in The value of the dimension; Then, make the following changes in each dimension of the random variable so that The probability of assigning values ​​to each point is close, as shown in formula (32): (32) ； in, is an indicator function that satisfies ; After transformation The sample point in The value of the dimension; For the The sample point in The value of the dimension; Finally, through Calculate The probability of assigning a representative point Then perform the following transformations: (33) ； After the above transformation, As the representative point set with small GF-deviation finally adopted; Furthermore, the probability space of the input random variables is divided into a set of non-overlapping representative regions ,Right now: (34) ； in, Indicates the The representative point of a Voronoi cell; Indicates the The representative point of a Voronoi cell; Indicates the Voronoi cells; Indicates the Voronoi cells; express dimensional Euclidean space; Step 4-2: Quantize the random function; According to the robust trajectory optimization problem shown in formula (30) obtained in step 3, when solving it, it is necessary to transform the objective function and constraint function containing random variables into deterministic statistics; Assume that the direct probability integral method is used to select Representative points, each representative point The corresponding distribution probability is , the mean and standard deviation of the objective function and constraint function are expressed as: (35) ； (36) ； in, represents the mean of the objective function; Indicates that the Representative points Substitute the obtained value of the objective function; Indicates the Representative points The corresponding distribution probability; represents the mean of the constraint function; represents the standard deviation of the constraint function; Indicates that the Representative points Substitute the obtained value of the constraint function.

10. The method for robust trajectory optimization of a tiltrotor aircraft based on direct probability integration method according to claim 9, characterized in that: The step 5 is specifically as follows: Substituting the mean of the objective function shown in formula (35) and the mean and standard deviation of the constraint function shown in formula (36) into formula (30), the robust trajectory optimization problem shown in formula (30) is transformed into a deterministic trajectory optimization problem that can be directly solved, as shown in formula (37): (37) ； At this point, the robust trajectory optimization problem of the tilt-rotor aircraft is transformed into a deterministic optimization problem. Finally, the deterministic trajectory optimization problem shown in formula (37) is solved to obtain the optimal control input and statistical characteristics of the state quantity of the tilt-rotor aircraft.

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