A Modular Reconfigurable Flight Array Dynamics Model and Fixed-Time Sliding Mode Control Method

Abstract

The present invention discloses a modular reconfigurable flight array dynamics model and a fixed-time sliding mode control method. The dynamics model includes establishing a coordinate system for describing the modular reconfigurable flight array; obtaining the inertia tensor of the modular reconfigurable flight array according to the established coordinate system; and establishing the dynamics model of the modular reconfigurable flight array according to the established coordinate system and the inertia tensor. According to the established coordinate system and the inertia tensor, the present invention constructs a dynamics model of the modular reconfigurable flight array. When the topological configuration of the flight array changes, there is no need to solve the centroid, thus realizing a fast modeling method. Further, the present invention adopts a variable exponential coefficient function, and the sliding mode controller designed based on this variable exponential coefficient function has the characteristics of fast convergence speed, non-singularity, and strong robustness.

Description

Technical Field

[0001] The present invention relates to a dynamic model of a modular reconfigurable flight array and a fixed-time sliding mode control method, belonging to the fields of unmanned rotorcraft design and control. Background Art

[0002] Due to advantages such as simple structure, portability, vertical takeoff and landing, and hovering in the air, multi-rotor aircraft have received increasing attention and research. Currently, the frontiers of rotorcraft research mainly include multi-motion mode unmanned aircraft, robot formation cooperation, and reconfigurable modular unmanned aircraft. Among them, the modular reconfigurable flight array changes its physical structure through self-assembly and self-separation between modules, so as to dynamically adapt to mission and environmental requirements.

[0003] Regarding the research on modular reconfigurable flight arrays, the literature "R. Oung, R D’Andrea. The distributed flight array: Design, implementation, and analysis of a modular vertical take-off and landing vehicle [J]. The International Journal of Robotics Research, 2014, 33(3): 375-400." proposed a modular flight array and studied its dynamic model and control. However, its modeling method requires knowing the centroid of the flight array. When the topological configuration of the flight array (the number of modules and the configuration of the flight array) changes, it is necessary to recalculate the centroid of the flight array, which increases the complexity of modeling and cannot achieve rapid modeling after the topological configuration changes.

[0004] There is less literature on the application of fixed-time sliding mode controllers in flight arrays. Existing fixed-time sliding mode control methods are all designed based on a constant exponential coefficient function y = -k 1 x p (k 1 , where p is a constant greater than zero), and this design method requires non-singularity processing and has a slow convergence speed. Summary of the Invention

[0005] The present invention provides a dynamic model of a modular reconfigurable flight array for establishing the dynamic model of a modular reconfigurable flight array, and further provides a fixed-time sliding mode control method for a modular reconfigurable flight array for fixed-time sliding mode control based on the established dynamic model of the modular reconfigurable flight array.

[0006] The technical solution of the present invention is as follows: A modular reconfigurable flight array dynamics model, including: establishing a coordinate system for describing the modular reconfigurable flight array; obtaining the inertia tensor of the modular reconfigurable flight array according to the established coordinate system; and establishing a modular reconfigurable flight array dynamics model according to the established coordinate system and the inertia tensor.

[0007] The coordinate system for describing the modular reconfigurable flight array includes:

[0008] Module coordinate system O M :{x M ,y M ,z M}; The module coordinate system takes the geometric center of the flight unit module as the coordinate origin, the z M axis is perpendicular to the flight unit module and points upward into the sky, and the coordinate system satisfies the right-hand rule; The module coordinate system is mainly used to express the moment of inertia J = diag(J Mx ,J My ,J Mz ) of the flight unit module, where J Mx is the moment of inertia of the flight unit module about the x M axis, J My is the moment of inertia of the flight unit module about the y M axis, and J Mz is the moment of inertia of the flight unit module about the z M axis;

[0009] Flight array coordinate system The flight array coordinate system takes any point in the flight array as the coordinate origin, and the directions of the coordinate axes are the same as those of the module coordinate system; The flight array coordinate system is mainly used to describe the position information of the geometric center of the i-th flight unit module in the flight array coordinate system and the angular velocities [p, q, r] rotating about the T axis respectively;

[0010] Inertial coordinate system O E :{x E ,y E ,z E}; The directions of the axes of the inertial coordinate system are the same as those of the axes of the flight array coordinate system; The inertial coordinate system is used to describe the position X E = [x E ,y E ,z E T of the modular reconfigurable flight array in three-dimensional space, and the attitude angles Θ = [φ, θ, ψ] T , where φ is the rotation about the x E ​The rotation angle of the axis, θ is the rotation angle about the y E axis, and ψ is the rotation angle about the z E axis.

[0011] Obtaining the inertia tensor of the modular reconfigurable flight array based on the established coordinate system includes: Based on the established coordinate system, applying the parallel axis theorem of moment of inertia, with the moment of inertia J = diag(J Mx , J My , J Mz ) of the flight unit module, the quantity n, and the mass m of a single flight unit module as the basis, calculating the inertia tensor of the modular reconfigurable flight array

[0012] wherein, is the position of the geometric center of the i-th flight unit module in the flight array coordinate system, J xx , J yy , J zz are the moments of inertia about the axis respectively, and J xy , J yx are the products of inertia with respect to the axis and the axis.

[0013] Establishing the dynamic model of the modular reconfigurable flight array includes: Based on the established coordinate system and inertia tensor, applying the Newton-Euler method to establish the dynamic model of the modular reconfigurable flight array; The dynamic model of the modular reconfigurable flight array includes the dynamic model of the translational motion of the modular reconfigurable flight array and the dynamic model of the rotational motion of the modular reconfigurable flight array;

[0014] The dynamic model of the translational motion of the modular reconfigurable flight array is as follows:

[0015]

[0016] In the formula, is the acceleration of the modular reconfigurable flight array along the x E axis, y E axis, and z E axis in the inertial coordinate system; g is the acceleration due to gravity; m is the mass of a single flight unit module; [φ, θ, ψ] T is the modular reconfigurable flight array rotating about the x E axis, y E axis, and z EAttitude angle of shaft rotation; cφ = cos(φ), sφ = sin(φ), cθ = cos(θ), sθ = sin(θ), cψ = cos(ψ), sψ = sin(ψ); T z Lift required for modular reconfigurable flight array flight

[0017] The dynamic model of the rotational motion of the modular reconfigurable flight array is as follows:

[0018]

[0019] In the formula, Are the angular accelerations about the x E axis, about the y E axis, and about the z E axis respectively; Are the angular velocities about the x E axis, about the y E axis, and about the z E axis respectively; J xx 、J yy 、J zz Are the moments of inertia about the axis respectively, and J xy 、J yx Are the products of inertia with respect to the axis and the axis; [M x ,M y ,M z T Are the driving torques about the x E axis, y E axis, and z E axis respectively;

[0020] The lift T z 、torque M x ,M y ,M z in the dynamic model are obtained by orthogonal decomposition of forces and torques, and the specific implementation form is as follows:

[0021]

[0022] Among them, the value of κ* is related to the rotation direction of the propeller: when the propeller rotates clockwise, κ* = 2; when the propeller rotates counterclockwise, κ* = 1; k F Is the propeller lift coefficient; k M Is the propeller torque coefficient; u i Is the actual control input of the actuator of the i-th flight unit module, that is, the pulse width; Respectively represent the position information of the geometric center of the i-th flight unit module in the flight array coordinate system.​

[0023] According to another aspect of the present invention, there is provided a modular reconfigurable flight array fixed-time sliding mode control method, including: extracting the control efficiency matrix B according to the lift and moment expressions described above e1 and the center-of-gravity compensation matrix B e2 ; establishing a control allocation model according to the control efficiency matrix B e1 and the center-of-gravity compensation matrix B e2 ; establishing an energy-optimal control allocation strategy according to the control allocation model, and mapping the virtual control input τ = [T z , M x , M y , M z T to each actuator in an energy-optimal relationship mapping

[0024] Extracting the control efficiency matrix B according to the lift and moment expressions e1 and the center-of-gravity compensation matrix B e2 , including:

[0025] Rewriting the lift T z , moment M x , M y , M z expressions into matrix form:

[0026]

[0027] According to the matrix forms of force and moment, the control efficiency matrix B z , M x , M y , M z T used to measure the relationship between the virtual control input [T 1 , u 2 , …, u n T and the actual control input of the actuator[u e1 is written as:

[0028]

[0029] And the center-of-gravity compensation matrix B z , M x , M y , M z T used to describe the relationship between the virtual control input [T e2 and the mass of a single flight unit module is written as:

[0030] ​​​​

[0031] In the formula, the value of κ* is related to the rotation direction of the propeller: when the propeller rotates clockwise, κ* = 2; when the propeller rotates counterclockwise, κ* = 1; k F is the lift coefficient of the propeller; k M is the torque coefficient of the propeller; u n is the actual control input of the actuator of the nth flight unit module; respectively represent the position information of the geometric center of the ith flight unit module in the flight array coordinate system, i = 1, 2,..., n; m 1 , m 2 , m n respectively represent the masses of the 1st, 2nd, and nth flight unit modules; g is the acceleration due to gravity.

[0032] Based on the control efficiency matrix B e1 and the center-of-gravity compensation matrix B e2 , a control allocation model is established as follows: In the form of the control efficiency matrix B e1 and the center-of-gravity compensation matrix B e2 , [T z , M x , M y , M z is mapped to each actuator, and the control allocation model is established as follows: T τ = B

[0033] τ = B e1 u + B e2 G

[0034] where τ = [T z , M x , M y , M z ; G = [m T g, m 1 g,..., m 2 g] n ; u = [u T , u 1 ,...u 2 ,...u n ; m n represents the mass of the nth flight unit module; g is the acceleration due to gravity; u n is the actual control input of the actuator of the nth flight unit module.

[0035] Based on the control allocation model, an energy-optimal control allocation strategy is established, and the virtual control input τ = [T z , M x , M y , Mz T Map to each actuator in an energy-optimal relationship, specifically: According to the established control allocation model, use the least squares method to establish an energy-optimal control allocation strategy, and map the virtual control input τ = [T z , M x , M y , M z T Map to each actuator in an energy-optimal relationship:

[0036]

[0037] s.t. τ = B e1 u + B e2 G

[0038] Solve the optimization equation to obtain:

[0039]

[0040] Among them, Ο(u) represents the objective function and is called the allocation strategy based on energy optimality; is the right generalized inverse matrix of matrix B e1 ; G = [m 1 g, m 2 g,..., m n g] T ; u = [u 1 , u 2 ,... u n ; m n represents the mass of the nth flight unit module; g is the acceleration due to gravity; u n is the actual control input of the actuator of the nth flight unit module.

[0041] The fixed-time sliding mode controller with variable exponential coefficients includes:

[0042] Establish the error dynamic equations of altitude and attitude angles according to the dynamic model:

[0043]

[0044] In the formula: In the formula, The lift T z , the moment M x , M y , M z ; are the altitude error, roll angle error, pitch angle error, and yaw angle error respectively, are the desired altitude, roll angle, pitch angle, and yaw angle; [h z ​​(t), h φ (t), h θ (t), h ψ (t)] T is a bounded unknown perturbation; f φ (φ, θ, ψ), f θ (φ, θ, ψ) and f ψ (φ, θ, ψ) represents a nonlinear function; g 11 , g 21 , g 22 , g 31 , g 32 and g 41 represent control gain functions;

[0045] Design a sliding mode surface in the following form based on the error equation:

[0046]

[0047] To analyze the dynamic characteristics of the sliding mode surface, taking the derivative gives:

[0048]

[0049] Establish a variable exponential coefficient sliding mode control law:

[0050]

[0051] According to the sliding mode control law and obtain the virtual control input τ = [T z , M x , M y , M z T ;

[0052] In the formula: s 1 , s 2 , s 3 , s 4 respectively represent the sliding mode surfaces for the control of altitude, roll angle, pitch angle, and yaw angle; is the hyperbolic tangent function, γ 11 , γ 12 , …, γ 41 , γ 42 > 0, κ 1 , κ 2 , κ 3 , κ 4 > 1 and 0 < ε 1 , ε 2 , ε 3 , ε 4 ​<1 is a constant; sin(*) is the sign function: when * > 0, sgn(*) = 1; when * = 0, sgn(*) = 0; when * < 0, sgn(*) = -1; k δ1 , k δ2 , k δ3 > 0 (δ = 1, 2, 3, 4) are control gains, λ δ > 0, μ δ > 0, p δ > 1 and satisfies λ δ > p δ μ δ .

[0053] The beneficial effects of the present invention are as follows: Based on the established coordinate system and inertia tensor, the present invention constructs a modular reconfigurable flight array dynamics model. When the topological configuration of the flight array changes, there is no need to solve the centroid, thus realizing a fast modeling method. Further, the present invention adopts a variable exponent coefficient function, and the sliding mode controller designed based on this variable exponent coefficient function has the characteristics of fast convergence speed, non-singularity, and strong robustness. BRIEF DESCRIPTION OF THE DRAWINGS

[0054] Figure 1 is the flowchart of the implementation of the modular reconfigurable flight array control;

[0055] Figure 2 is the schematic diagram of each coordinate of the modular reconfigurable flight array;

[0056] Figure 3 is the schematic diagram of the flight unit module structure;

[0057] Figure 4 is the actuator of the modular reconfigurable flight array;

[0058] Figure 5 is the height and attitude control block diagram of the modular reconfigurable flight array;

[0059] Figure 6 is the configuration schematic diagram of the "X"-type six-module flight array;

[0060] Figure 7 is the error convergence curve graph under different initial states;

[0061] Figure 8 is Figure 6 the height and attitude tracking simulation diagram of the flight array shown in DETAILED DESCRIPTION OF THE EMBODIMENTS

[0062] The following will further illustrate the invention in conjunction with the drawings and embodiments, but the content of the present invention is not limited to the described scope.

[0063] Embodiment 1: AsFigures 1-8 As shown in the figure, a modular reconfigurable flight array dynamics model includes: establishing a coordinate system for describing the modular reconfigurable flight array; obtaining the inertia tensor of the modular reconfigurable flight array based on the established coordinate system; and establishing a modular reconfigurable flight array dynamics model based on the established coordinate system and inertia tensor.

[0064] Further, the coordinate system for describing the modular reconfigurable flight array includes:

[0065] Step 1.1: Establish a coordinate system (as shown in the figure) for describing the spatial position and attitude of the modular reconfigurable flight array. The coordinate system includes: Figure 2 As shown in the figure, the coordinate system includes:

[0066] 1) Module coordinate system O M :{x M ,y M ,z M}; The module coordinate system takes the geometric center of the flight unit module as the coordinate origin. The z M axis is perpendicular to the flight unit module and points upward. The coordinate system satisfies the right-hand rule. The module coordinate system is mainly used to express the moment of inertia J = diag(J Mx ,J My ,J Mz ) of the flight unit module, where J Mx is the moment of inertia of the flight unit module about the x M axis, J My is the moment of inertia of the flight unit module about the y M axis, and J Mz is the moment of inertia of the flight unit module about the z M axis. It should be noted that the module coordinate system of each flight unit module is constructed according to the above establishment process, and it is ensured that the axis directions of the module coordinate systems of each flight unit module are consistent;

[0067] 2) Flight array coordinate system The flight array coordinate system takes any point in the flight array as the coordinate origin (usually selects the geometric center of any flight unit module / or selects the geometric center of the entire flight array as the coordinate origin). The directions of the coordinate axes are the same as those of the module coordinate system. The flight array coordinate system is mainly used to describe the position information of the i-th flight unit module in the flight array coordinate system and the angular velocities [p, q, r] T of rotation about the

[0068] 3) Inertial coordinate system O E :{x E ,yE , z E}; The directions of the axes of the inertial coordinate system are consistent with those of the flight array coordinate system; the inertial coordinate system is used to describe the position X of the modular reconfigurable flight array in three-dimensional space E = [x E , y E , z E T , and the attitude angle Θ = [φ, θ, ψ] T , where φ is the rotation angle about the x E axis (roll angle), θ is the rotation angle about the y E axis (pitch angle), and ψ is the rotation angle about the z E (yaw angle);

[0069] The modular reconfigurable flight array is assembled by splicing multiple flight unit modules. For illustrative purposes rather than limitation, some configurations are shown in the accompanying drawings of the present invention, such as Figure 2 showing a modular reconfigurable flight array constructed from 9 regular hexagonal flight unit modules, Figure 3 giving Figure 2 the corresponding schematic diagram of the flight unit module structure in Figure 3 giving the schematic diagram of the corresponding actuator; Figure 6 is another topological structure sketch.

[0070] Further, obtaining the inertia tensor of the modular reconfigurable flight array according to the established coordinate system includes:

[0071] Step 1.2. According to the coordinate system established in Step 1.1, using the parallel axis theorem of moment of inertia, based on the moment of inertia J = diag(J Mx , J My , J Mz ) of the flight unit module, the number n, and the mass m of a single flight unit module, calculate the inertia tensor of the modular reconfigurable flight array

[0072]

[0073] Among them, is the position of the geometric center of the i-th flight unit module in the flight array coordinate system, J xx , J yy , J zz are the moments of inertia about the axes, and J xy , J yx are relative to the axis and ​Product of inertia of the shaft; the masses of the individual flight unit modules in the modular reconfigurable flight array are the same.

[0074] Furthermore, based on the established coordinate system and inertia tensor Establish a dynamic model of the modular reconfigurable flight array, including:

[0075] Step 1.3: According to the coordinate system established in Step 1.1 and the inertia tensor in Step 1.2 Use the Newton-Euler method to establish the dynamic model of the modular reconfigurable flight array:

[0076] The dynamic model of the translational motion of the modular reconfigurable flight array is:

[0077]

[0078] Where is the acceleration of the modular reconfigurable flight array along the x E axis, y E axis, and z E axis in the inertial coordinate system; g is the acceleration due to gravity; [φ, θ, ψ] T are the attitude angles of the modular reconfigurable flight array rotating about the x E axis, y E axis, and z E axis respectively; cφ = cos(φ), sφ = sin(φ), cθ = cos(θ), sθ = sin(θ), cψ = cos(ψ), sψ = sin(ψ); T z is the lift required for the modular reconfigurable flight array to fly, which is generated by the high-speed rotation of the propeller;

[0079] The dynamic model of the rotational motion of the modular reconfigurable flight array is:

[0080]

[0081] Where are the angular accelerations about the x E axis, about the y E axis, and about the z E axis respectively; are the angular velocities about the x E axis, about the y E axis, and about the z E axis respectively; [M x , M y , M z T are the torques about the x E axis, y E axis, and z E axis respectively; ​

[0082] The lift force T in the kinetic model z and the moment M x , M y , M z , which are obtained by the orthogonal decomposition of the force and moment, and the specific implementation form is as follows:

[0083]

[0084] Among them, the value of κ* is related to the rotation direction of the propeller: when the propeller rotates clockwise, κ* = 2; when the propeller rotates counterclockwise, κ* = 1; k F is the lift coefficient of the propeller; k M is the torque coefficient of the propeller; u i is the actual control input of the actuator of the i-th flight unit module, that is, the pulse width.

[0085] According to another aspect of the present invention, there is provided a modular reconfigurable flight array fixed-time sliding mode control method, including: extracting the control efficiency matrix B e1 and the center-of-gravity compensation matrix B e2 according to the above lift and moment expressions; establishing a control allocation model based on the control efficiency matrix B e1 and the center-of-gravity compensation matrix B e2 ; establishing an energy-optimal control allocation strategy based on the control allocation model, and mapping the virtual control input τ = [T z , M x , M y , M z T to each actuator in an energy-optimal relationship mapping.

[0086] Furthermore, extracting the control efficiency matrix B e1 and the center-of-gravity compensation matrix B e2 according to the lift and moment expressions, including:

[0087] Step 2. Extract the control efficiency matrix B e1 and the center-of-gravity compensation matrix B e2 according to the force and moment expressions in Step 1; the specific implementation steps are as follows:

[0088] Step 2.1. Rewrite the force and moment expressions into matrix form:

[0089]

[0090] According to the matrix form of the force and moment, the virtual control input [T z , M x , M​y , M z T and the actual control input [u 1 , u 2 , …, u n T of the control efficiency matrix B that describes the relationship between e1 is written as:

[0091]

[0092] and the centroid compensation matrix B used to describe the virtual control input [T z , M x , M y , M z T and the mass m of a single flight unit module is written as: e2 is written as:

[0093]

[0094] In the formula, the value of κ* is related to the rotation direction of the propeller: when the propeller rotates clockwise, κ* = 2; when the propeller rotates counterclockwise, κ* = 1; k F is the propeller lift coefficient; k M is the propeller torque coefficient; u n is the actual control input of the actuator of the nth flight unit module; respectively represent the position information of the geometric center of the ith flight unit module in the flight array coordinate system, i = 1, 2,..., n; m 1 , m 2 , m n respectively represent the masses of the 1st, 2nd, and nth flight unit modules; g is the acceleration due to gravity.

[0095] Furthermore, based on the control efficiency matrix B e1 and the centroid compensation matrix B e2 , a control allocation model is established, specifically:

[0096] Step 2.2, through the control efficiency matrix B e1 and the centroid compensation matrix B e2 , a mathematical model for describing the control allocation problem is established, that is, [T e1 is mapped to each actuator in the form of the matrix B z , M x , M y , M z T to establish the following control allocation model:

[0097] τ = B e1 ​​​​u + B e2 G

[0098] where τ = [T z , M x , M y , M z T , G = [m 1 g, m 2 g, …, m n g] T ; u = [u 1 , u 2 ,... u n .

[0099] Furthermore, based on the control allocation model, an energy - optimal control allocation strategy is established, mapping the virtual control input τ = [T z , M x , M y , M z T produced by a fixed - time sliding - mode controller with variable exponential coefficients to each actuator in an energy - optimal relationship, specifically:

[0100] Step 3: Based on the control allocation model established in Step 2, the least - squares method is used to establish an energy - optimal control allocation strategy, mapping the virtual control input τ = [T z , M x , M y , M z T to each actuator in an energy - optimal relationship:

[0101]

[0102] s.t. τ = B e1 u + B e2 G

[0103] Solve the optimization equation to obtain:

[0104]

[0105] where Ο(u) represents the objective function, is the right - generalized inverse matrix of matrix B e1 ; is called the allocation strategy based on energy optimization.

[0106] Steps 1, 2, and 3 complete the establishment of the modular reconfigurable flight array dynamics model and the design of the control allocation strategy, laying a preliminary foundation for describing the control of the modular reconfigurable flight array; in the following text, we will elaborate on the controller design of the modular reconfigurable flight array in detail:​​​

[0107] Furthermore, the fixed-time sliding mode controller with variable exponential coefficients includes:

[0108] Step 4: According to the dynamic model in Step 1, the present invention proposes a fixed-time sliding mode controller with variable exponential coefficients for the height and attitude angle control of a modular reconfigurable flight array. The specific implementation steps are as follows:

[0109] Step 4.1: Establish the error dynamic equations for height and attitude angles based on the dynamic model:

[0110]

[0111] Where, are the height error, roll angle error, pitch angle error, and yaw angle error respectively, is the desired height, roll angle, pitch angle, and yaw angle; [h z (t), h φ (t), h θ (t), h ψ (t)] T is a bounded unknown disturbance;

[0112] The expressions of the nonlinear functions f φ (φ, θ, ψ), f θ (φ, θ, ψ), and f ψ (φ, θ, ψ) are as follows:

[0113]

[0114] The expressions of the control gain functions g 11 , g 21 , g 22 , g 31 , g 32 , and g 41 are:

[0115]

[0116] Step 4.2: Design a sliding mode surface in the following form based on the error equation:

[0117]

[0118] Where, is the hyperbolic tangent function, γ 11 , γ 12 , …, γ 41 , γ 42 > 0, κ 1 , κ 2 , κ3 , κ 4 > 1 and 0 < ε 1 , ε 2 , ε 3 , ε 4 < 1 is a constant; sin(*) is the sign function: when * > 0, sgn(*) = 1; when * = 0, sgn(*) = 0; when * < 0, sgn(*) = -1; s 1 , s 2 , s 3 , s 4 respectively represent the sliding mode surfaces for the control of altitude, roll angle, pitch angle, and yaw angle;

[0119] To analyze the dynamic characteristics of the sliding mode surface, taking the derivative gives:

[0120]

[0121] Step 4.3: To make the sliding mode surface tend to zero, a new variable - exponent - coefficient sliding - mode control law is proposed in the present invention:

[0122]

[0123] where, k δ1 , k δ2 , k δ3 > 0 (δ = 1, 2, 3, 4) are control gains, λ δ > 0, μ δ > 0, p δ > 1 and satisfy λ δ > p δ μ δ ;

[0124] According to the sliding - mode control law and obtain the virtual control input τ = [T z , M x , M y , M z T .

[0125] Furthermore, the following is given:

[0126] Step 5: According to the fixed - time sliding - mode controller designed in Step 4, in this step, the stability of the controller τ 1 is analyzed and described. The stability analysis of the controller τ 2 , τ 3 , τ 4 is the same as that of τ 1 . The specific implementation steps include:

[0127] ​Step 5.1: Substitute the sliding mode control law into the derivative of the sliding mode surface, and the sliding mode reaching law can be obtained:

[0128]

[0129] Step 5.2: Define the Lyapunov function with respect to the sliding mode surface s 1 :

[0130]

[0131] Take the derivative of the Lyapunov function and combine it with the sliding mode reaching law to get:

[0132]

[0133] It can be seen from the derivative of the Lyapunov function that the robustness of the sliding mode controller is mainly reflected by . When we select the parameter k 13 greater than the upper bound of the disturbance , it can surely ensure that always holds; therefore, the controller proposed by the present invention has strong robustness.

[0134] Step 5.3: According to the derivative expression of the Lyapunov function obtained in Step 5.2, this part will conduct fixed-time stability analysis on the reaching law; rewrite the derivative of the Lyapunov function into the following form:

[0135]

[0136] Integrate both ends to get:

[0137]

[0138] Let Therefore, the definite integral can be written in the following inequality form:

[0139]

[0140] It can be seen from the inequality that is a strictly monotonically increasing function of V(t). When and only when V(t) = 0, the function Therefore, define the following limit expression:

[0141]

[0142] where T(s 1 (0)) is the settling time, that is, the time required for the sliding mode surface s 1 to converge to the sliding surface S = 0;

[0143] When holds, Therefore, T(s 1 (0)) can be rewritten as:

[0144]

[0145] T(s 1 (0))'s upper bound in time can be defined by the following generalized definite integral:

[0146]

[0147] By solving the generalized definite integral, the upper bound in time is defined as:

[0148]

[0149] That is, the sliding mode surface s 1 converges to the sliding surface S = 0 within time T max ; From the upper bound of the convergence time, it can be seen that its upper bound in time is independent of the initial conditions, that is, regardless of the initial value, it will converge to zero within time T max .

[0150] To verify the effectiveness of the modeling method and the fixed-time sliding mode controller design in the present invention, the present invention conducts simulation verification on the fixed convergence of the sliding mode reaching law and the effectiveness of the controller. Figure 7 shows the convergence characteristics of the sliding mode reaching law under different initial values of 5, 10, 15, and 20. The control parameters are selected as k 11 = 2, k 12 = 6, k 13 = 2, p 1 = 1.2, λ 1 = 0.2, μ = 0.1, and d = sin(10t) (d z = 1), and the convergence time can be calculated as T max = 1.489 seconds from the upper bound in time. This shows that the control in the present invention still has the fixed-time convergence characteristic in the presence of external disturbances.

[0151] For the controller, select the six-module reconfigurable flight array shown in Figure 6 for simulation verification:

[0152] Figure 3 shows the structure of a flight unit module in the flight array, and its parameters are selected as shown in Table 1.

[0153] Table 1 Flight unit module parameter table

[0154]

[0155] In as Figure 6In the flight array coordinate system shown, the coordinate information of each flight unit module is shown in Table 2.

[0156] Table 2 Figure 6 Coordinate Information of Each Module of the Modular Reconfigurable Flight Array Shown (Unit: meter) Table 2-1

[0157]

[0158] Table 2-2

[0159]

[0160] The selection of controller parameters is shown in Table 3:

[0161] Table 3

[0162]

[0163] The specific implementation block diagram of flight array control is as Figure 5 shown. This control framework is applicable to the height and attitude control problems of flight arrays of all configurations. Based on the dynamic model and control method proposed in the present invention, for Figure 6 the six-module flight array shown, height and attitude tracking simulations are carried out. The control parameters are selected as the parameters shown in Table 3, and the simulation results are as Figure 8 shown. Figure 8 shows the trajectory tracking curves of height, roll angle, pitch angle, and yaw angle. It can be seen from the tracking curves that the modeling method and control method proposed in the present invention have good effects.

[0164] The specific embodiments of the present invention have been described in detail above with reference to the accompanying drawings. However, the present invention is not limited to the above embodiments. Within the knowledge scope of those of ordinary skill in the art, various changes can be made without departing from the spirit of the present invention.

Claims

1. A dynamic model of a modular reconfigurable flight array, characterized in that, it includes: Establish a coordinate system for describing the modular reconfigurable flight array; According to the established coordinate system, obtain the inertia tensor of the modular reconfigurable flight array; According to the established coordinate system and inertia tensor, establish a dynamic model of the modular reconfigurable flight array; The inertial tensor of the modular reconfigurable flight array obtained based on the established coordinate system includes: Based on the established coordinate system, using the parallel axis theorem of the moment of inertia, with the moment of inertia J = diag(J Mx , J My , J Mz ) of the flight unit module, the quantity n, and the mass m of a single flight unit module as the basis, calculate the inertial tensor of the modular reconfigurable flight array Among them, is the position of the geometric center of the i-th flight unit module in the flight array coordinate system, J xx , J yy , J zz are the moments of inertia about the axis respectively, and J xy , J yx are the products of inertia with respect to the axis and the axis.

2. The dynamic model of the modular reconfigurable flight array according to claim 1, characterized in that, the coordinate system for describing the modular reconfigurable flight array includes: Module coordinate system O M :{x M , y M , z M}; The module coordinate system takes the geometric center of the flight unit module as the coordinate origin, and the z M axis is perpendicular to the flight unit module and points upward. The coordinate system satisfies the right-hand rule. The module coordinate system is mainly used to represent the moment of inertia of the flight unit module J = diag(J M x, J M y, J M z), where J M x is the moment of inertia of the flight unit module about the x M axis, J My is the moment of inertia of the flight unit module about the y M axis, and J Mz is the moment of inertia of the flight unit module about the z M axis; Flight array coordinate system The flight array coordinate system takes any point in the flight array as the coordinate origin, and the axes have the same directions as those of the module coordinate system; the flight array coordinate system is mainly used to describe the position information of the geometric center of the i-th flight unit module in the flight array coordinate system and the angular velocities [p, q, r] of rotation about the axes respectively T ; Inertial coordinate system O E :{x E ,y E ,z E}; The directions of the axes of the inertial coordinate system are consistent with those of the flight array coordinate system; The inertial coordinate system is used to describe the position X of the modular reconfigurable flight array in three-dimensional space E =[x E ,y E ,z E T , and the attitude angle Θ = [φ, θ, ψ] T , where φ is the rotation angle about the x E axis, θ is the rotation angle about the y E axis, and ψ is the rotation angle about the z E axis.​ 3. The dynamic model of the modular reconfigurable flight array according to claim 1, characterized in that, The established coordinate system and inertia tensor Establish a dynamic model of a modular reconfigurable flight array, including: establishing a dynamic model of a modular reconfigurable flight array using the Newton-Euler method based on the established coordinate system and inertia tensor; the dynamic model of a modular reconfigurable flight array includes a dynamic model of the translational motion of a modular reconfigurable flight array and a dynamic model of the rotational motion of a modular reconfigurable flight array; the dynamic model of the translational motion of the modular reconfigurable flight array is: In the formula, is the acceleration of the modular reconfigurable flight array along the x E axis, y E axis, and z E axis; g is the acceleration due to gravity; m is the mass of a single flight unit module; [φ, θ, ψ] T are the attitude angles of the modular reconfigurable flight array rotating about the x E axis, y E axis, and z E axis respectively; cφ = cos(φ), sφ = sin(φ), cθ = cos(θ), sθ = sin(θ), cψ = cos(ψ), sψ = sin(ψ); T z is the lift required for the modular reconfigurable flight array to fly; the dynamic model of the rotational motion of the modular reconfigurable flight array is: In the formula, are the angular accelerations of rotation about the x E axis, about the y E axis, and about the z E axis respectively; are the angular velocities of rotation about the x E axis, about the y E axis, and about the z E axis respectively; J xx , J yy , J zz are the moments of inertia about the axis respectively, and J xy , J yx are the products of inertia with respect to the axis and the axis; [M x , M y , M z T are the driving torques about the x E axis, y E axis, and z E axis respectively;​ Lift T in the kinetic model z , moment M x , M y , M z , obtained by orthogonal decomposition of force and moment, and the specific implementation form is as follows: Among them, the value of κ* is related to the rotation direction of the propeller: when the propeller rotates clockwise, κ* = 2; when the propeller rotates counterclockwise, κ* = 1; k F is the lift coefficient of the propeller; k M is the torque coefficient of the propeller; u i is the actual control input of the actuator of the i-th flight unit module, that is, the pulse width; respectively represent the position information of the geometric center of the i-th flight unit module in the flight array coordinate system.

4. A fixed-time sliding mode control method for a modular reconfigurable flight array, characterized in that, it includes: Extract the control efficiency matrix B according to the lift and moment expressions described in claim 3 e1 and the center of gravity compensation matrix B e2 ; Based on the control efficiency matrix B e1 and the centroid compensation matrix B e2 , a control allocation model is established; According to the control allocation model, an energy-optimal control allocation strategy is established to map the virtual control input τ = [T z , M x , M y , M z generated by a fixed-time sliding mode controller with a variable exponential coefficient T to each actuator in an energy-optimal relationship mapping.

5. The fixed-time sliding mode control method for a modular reconfigurable flight array according to claim 4, characterized in that, Extract the control efficiency matrix B based on the lift and moment expressions e1 and the center of gravity compensation matrix B e2 , including: The lift force T z , the moment M x , M y , M z Rewrite the expression into matrix form: According to the matrix forms of force and moment, the control efficiency matrix B z , M x , M y , M z T used to measure the virtual control input [T 1 , u 2 , …, u n T and the actual control input [u e1 of the actuator is written as:​​ and for describing the virtual control input [T z , M x , M y , M z T and the center-of-gravity compensation matrix B of the mass of a single flight unit module e2 is written as:​ In the formula, the value of κ* is related to the rotation direction of the propeller: when the propeller rotates clockwise, κ* = 2; when the propeller rotates counterclockwise, κ* = 1; k F is the lift coefficient of the propeller; k M is the torque coefficient of the propeller; u n is the actual control input of the actuator of the nth flight unit module; respectively represent the position information of the geometric center of the ith flight unit module in the flight array coordinate system, i = 1, 2,..., n; m 1 , m 2 , m n respectively represent the masses of the 1st, 2nd, and nth flight unit modules; g is the acceleration due to gravity.

6. The fixed-time sliding mode control method for a modular reconfigurable flight array according to claim 4, characterized in that, According to the control efficiency matrix B e1 and the centroid compensation matrix B e2 , a control allocation model is established as follows: Taking the control efficiency matrix B e1 and the centroid compensation matrix B e2 , map [T z , M x , M y , M z T to each actuator, and establish the control allocation model as follows: τ = B e1 u + B e2 G where τ = [T z , M x , M y , M z T , G = [m 1 g, m 2 g, …, m n g] T ; u = [u 1 , u 2 ,... u n ; m n represents the mass of the nth flight unit module; g is the acceleration due to gravity; u n is the actual control input of the actuator of the nth flight unit module.​ 7. The fixed-time sliding mode control method for a modular reconfigurable flight array according to claim 4, characterized in that, According to the control allocation model, an energy-optimal control allocation strategy is established, which converts the virtual control input τ generated by the fixed-time sliding mode controller with variable exponential coefficient into [T z ,M x ,M y ,M z ] T The energy-optimal relationship is mapped to each actuator. Specifically, based on the established control allocation model, an energy-optimal control allocation strategy is established using the least squares method. The virtual control input τ generated by the fixed-time sliding mode controller with a variable exponential coefficient is z ,M x ,M y ,M z ] T Mapped to each actuator in an energy-optimal relationship: such that τ = B e1 u + B e2 G Solve the optimization equation to obtain: Among them, Ο(u) represents the objective function, which is called the energy-optimal-based allocation strategy; is the right generalized inverse matrix of matrix B e1 ; G = [m 1 g, m 2 g,..., m n g]; T ; u = [u 1 , u 2 ,... u n ; m n represents the mass of the nth flight unit module; g is the acceleration due to gravity; u n is the actual control input of the actuator of the nth flight unit module.

8. The fixed-time sliding mode control method for a modular reconfigurable flight array according to claim 4, characterized in that, the fixed-time sliding mode controller with variable exponential coefficients includes: Establish an error dynamic equation of height and attitude angle according to the dynamic model: In the formula: Lift force T z , Moment M x , M y , M z ; Are the altitude error, roll angle error, pitch angle error, and yaw angle error respectively, Are the desired altitude, roll angle, pitch angle, and yaw angle; [h z (t), h φ (t), h θ (t), h ψ (t)] T Are bounded unknown disturbances; f φ (φ, θ, ψ), f θ (φ, θ, ψ) and f ψ (φ, θ, ψ) represent nonlinear functions; g 11 , g 21 , g 22 , g 31 , g 32 And g 41 Represent control gain functions; Design a sliding mode surface in the following form based on the error equation: In order to analyze the dynamic characteristics of the sliding mode surface, the derivative can be obtained: Establish a variable exponential coefficient sliding mode control law: According to the sliding mode control law and obtain the virtual control input τ = [T z , M x , M y , M z T ;​ where: s 1 , s 2 , s 3 , s 4 respectively represent the sliding mode surfaces for height, roll angle, pitch angle, and yaw angle control; is the hyperbolic tangent function, γ 11 , γ 12 , …, γ 41 , γ 42 > 0, κ 1 , κ 2 , κ 3 , κ 4 > 1 and 0 < ε 1 , ε 2 , ε 3 , ε 4 < 1 are constants; sin(*) is the sign function: when * > 0, sgn(*) = 1; when * = 0, sgn(*) = 0; when * < 0, sgn(*) = -1; k δ1 , k δ2 , k δ3 > 0 (δ = 1, 2, 3, 4) is the control gain, λ δ > 0, μ δ > 0, p δ > 1 and satisfies λ δ > p δ μ δ .

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