Sliding mode control method for quadrotor suspension system based on extended state observer

Abstract

A sliding mode control method for a quadrotor suspension system based on an extended state observer (ESO) includes: constructing a dynamic model of the quadrotor suspension system; analyzing the established dynamic model to obtain virtual control torques corresponding to each state variable of the quadrotor suspension system, and decomposing the entire quadrotor suspension system into multiple subsystems based on these virtual control torques; treating various uncertain perturbations in the quadrotor suspension system as extended state variables of the system, and designing a nonlinear extended state observer based on these variables for real-time estimation of the system's perturbations; and designing a nonsingular fast sliding mode controller for each subsystem based on the perturbation estimates provided by the nonlinear extended state observer. This method enables the quadrotor suspension system to accurately track the desired target while effectively and rapidly suppressing load swing.

Description

Technical Field

[0001] The present invention relates to the field of automatic control of the posture of a quadrotor suspension system, and in particular to a sliding mode control method of a quadrotor suspension system based on an extended state observer. Background Art

[0002] Amidst the booming low-altitude economy, the application of various drones is expanding. Quadcopters are emerging as a leading trend in aerial transportation. Traditional ground transportation relies on vehicles, which are limited by road conditions and terrain. This significantly reduces efficiency, particularly in mountainous areas and densely populated urban areas with dense skyscrapers. Traditional aerial transportation relies primarily on helicopters, which are costly. In contrast, quadcopters, with their ability to suspend loads via tethers, significantly reduce their dependence on terrain. Their flight altitude cleverly bridges the gap between ground transportation and the flight paths of large passenger aircraft, avoiding the use of existing transportation resources. Compared to helicopters, quadcopters offer unique advantages in suspended transport. Their small size, lack of specialized takeoff and landing facilities, and increased flexibility make them particularly suitable for transporting cargo between high-rise buildings in cities, offering a high cost-effectiveness. However, during suspended transport, the inertia of the load often causes unavoidable swaying of the quadcopter. This not only impairs the trajectory tracking capabilities of the transport system but also poses significant safety risks. Therefore, an important current research direction is to explore how to effectively control the quadrotor suspension system to overcome this challenge.

[0003] Most of the control strategies currently used in quadrotor suspension systems have many limitations. They generally tend to treat the load as a disturbance factor and rely on disturbance suppression methods to reduce the swing angle oscillation. However, given the wide range and diversity of load types, simply treating the load as a disturbance factor is not enough.

[0004] 8 / The strategy of simply treating the load as a disturbance without deeply exploring the intrinsic impact mechanism of load swing on the dynamic behavior of the system is obviously flawed. Summary of the Invention

[0005] To solve the above technical problems, the present invention provides a sliding mode control method for a quadrotor suspension system based on an extended state observer. This method enables the quadrotor suspension system to effectively and quickly suppress the swing of the load while accurately tracking the desired target.

[0006] The technical solution adopted by the present invention is:

[0007] The sliding mode control method of a quadrotor suspension system based on an extended state observer comprises the following steps:

[0008] Step 1: Construct a dynamic model of the quadrotor suspension system;

[0009] Step 2: Analyze the dynamic model of the quadrotor suspension system established in step 1 to obtain the virtual control torque corresponding to each state variable of the quadrotor suspension system. Based on this, decompose the entire quadrotor suspension system into multiple subsystems.

[0010] Step 3: Consider the various uncertain disturbances in the quadrotor suspension system as the system's extended state variables. Based on this, a nonlinear extended state observer is designed to estimate the system's disturbances in real time.

[0011] Step 4: Design a nonsingular fast sliding mode controller for each subsystem based on the disturbance estimates provided by the nonlinear extended state observer.

[0012] In step 1, the quadrotor suspension system includes a quadrotor drone, a suspension rope used to suspend a load, and a suspended load. The modeling process of the dynamic model of the quadrotor suspension system includes:

[0013] By using the Lagrange equation and combining the relationship between the coordinates of the center of mass of the quadrotor drone and the center of mass of the suspended load, the system position swing angle dynamic model is obtained to describe the position of the load in space and its swing characteristics.

[0014] According to the Euler equation, the relationship between the rotational torque around the axis and the attitude angular acceleration is obtained, and then the system attitude dynamics model is determined to reflect the attitude changes of the quadrotor suspension system during flight.

[0015] Based on the inertial coordinate system, the system position swing angle dynamic model is established. It is assumed that the center of mass position of the quadrotor drone is expressed as (x, y, z) in the inertial coordinate system, and the load position is expressed as (x p ,y p ,z p ),like Figure 1 The relative relationship between the coordinates of the center of mass of the payload and the coordinates of the center of mass of the quadrotor drone is:

[0016]

[0017] Where: L is the length of the hanging rope; α is the swing angle projection on the yz plane, and β is the swing angle projection on the xz plane;

[0018] The system position swing angle dynamic model established using the Lagrange equation is as follows:

[0019]

[0020] Among them, q=[x, y, z, α, β, φ, θ, ψ] T, [·] T is the transpose of the vector [·], is the first-order derivative of q with respect to time, is the second-order derivative of q with respect to time, u is the driving force of the system in the position swing subsystem, which is the lift torque u1 of the quadrotor suspension system here.

[0021] M(q), G(q) and b are the model parameter matrices defined as follows:

[0022]

[0023] In the above formula, m 11 、m 15 、m 22 、m 24 、m 25 、m 33 、m 34 、m 35 、m 42 、m 43 、m 44 、m 51 、m 52 、m 53 、m 55 They are all model parameters, and the specific expressions are as follows:

[0024] m 11 =m 22 =m 33 =M+m,m 15 =mLcos(β)

[0025] m 24 =mLcos(β)cos(α),m 25 =-mLsin(β)sin(α)

[0026] m 34 =mLcos(β)sin(α),m 35 =mLsin(β)cos(α)

[0027] m 42 =cos(α),m 43 =sin(α),m 44 =cos(β)

[0028] m 51 =cos(β),m 52 =-sin(β)sin(α),m 53 =sin(β)cos(α),m 55 =L

[0029] Where: M and m are the masses of the UAV and the suspended load respectively.

[0030]

[0031] In the above formula, C 15 、C 25 、C 35 、C 24 、C 34 、C 44 、C 54 They are all model parameters, and the specific expressions are as follows:

[0032]

[0033] G(q)=[00g sin(α)g cos(α)sin(β)g] T ;

[0034] b=[b1 b2 b3 00] T ;

[0035] In the above formula, g is the acceleration due to gravity. b1, b2, and b3 are the transformation matrices from the drone's body coordinate system to the inertial coordinate system. The specific expressions are as follows:

[0036] b1=cos(φ)sin(θ)cos(ψ)+sin(φ)sin(ψ)

[0037] b2=cos(φ)sin(θ)sin(ψ)-sin(φ)cos(ψ)

[0038] b3=cos(φ)cos(θ)

[0039] The system attitude dynamics model established using the Euler equation is as follows:

[0040]

[0041] Where l is the wheelbase of the quadrotor drone; φ, θ, ψ are the angles of the drone’s counterclockwise rotation around the body’s coordinate axis; are the second-order derivatives of φ, θ, and ψ with respect to time; are the first-order derivatives of φ, θ, and ψ with respect to time; u φ ,u θ ,u ψ They represent the counterclockwise rotation torque of the drone around the body coordinate axis; I x ,I y ,I z They represent the moment of inertia of the UAV rotating counterclockwise around the body coordinate axis; d φ ,d θ ,dψ is the uncertainty disturbance of the system.

[0042] The four-rotor suspension system adopts a cross structure. The four rotor speeds ω1, ω2, ω3, ω4 are related to the system lift torque u1 and the torque around the three axes u φ ,u θ ,u ψ The corresponding relationship is:

[0043]

[0044] Among them, c T is the lift coefficient, c M is the counter torque coefficient.

[0045] In step 2, first, set the virtual control torque u x ,u y ,u z :

[0046]

[0047] Where M represents the mass of the drone and m represents the mass of the payload.

[0048] Then the virtual control torque u can be obtained x ,u y ,u z With the system lift torque u1 and the desired attitude angle φ d ,θ d The relationship between:

[0049]

[0050] The position and swing angle dynamics model of the quadrotor suspension system can be expressed as:

[0051]

[0052] in, Represent the second-order derivatives of x, y, and z with respect to time, are the second and first order derivatives of α and β with respect to time, respectively, d x ,d y ,d z ,d α ,d β represents the uncertainty disturbance of the system, and g represents the acceleration of gravity.

[0053] will u x As the main control quantity of the β subsystem, u y As the main control quantity of the α subsystem, u x Decompose to obtain the control quantity u of the x subsystem x1and the control quantity u of the β subsystem x2 ,u y Decomposed into the control quantity u of the y subsystem y1 and the control quantity u of the α subsystem y2 , thus we get u x 、u y The expression:

[0054]

[0055] Let P = [x, y] T , Q=[α,β] T ,Ω=[z,φ,θ,ψ] T , the entire quadrotor suspension system can be redefined as:

[0056]

[0057] in, is the second-order derivative of P, Q, and Ω with respect to time; P, Q, and Ω represent the state variable vectors of the translation subsystem, the yaw subsystem, and the altitude and attitude subsystem, respectively; f(P), f(Q), and f(Ω) are the model variables in the translation subsystem model, the yaw subsystem model, and the altitude and attitude subsystem model, excluding the control variables; g(P), g(Q), and g(Ω) are the control variable coefficients in the translation subsystem model, the yaw subsystem model, and the altitude and attitude subsystem model, respectively. f(P) = [f(x), f(y)] T ,f(Q)=[f(α),f(β)] T , f(Ω)=[f(z),f(φ),f(θ),f(ψ)] T , f(x), f(y), f(α), f(β), f(z), f(φ), f(θ), and f(ψ) represent the model variables in the x, y, α, β, φ, θ, and ψ subsystem models, respectively, except for the items where the control quantities are located.

[0058] g(P)=[g(x),g(y)] T , g(Q)=[g(α),g(β)] T , g(Ω)=[g(z),g(φ),g(θ),g(ψ)] T , g(x), g(y), g(α), g(β), g(z), g(φ), g(θ), and g(ψ) represent the control coefficients of the x, y, α, β, φ, θ, and ψ subsystem models, respectively.

[0059] u P 、u Q 、u Ω They are the control quantities of translation subsystem, swing subsystem, height and attitude subsystem respectively.

[0060] u P =[u x1 ,u y2 ] T ,u Q =[u y2 ,u x2 ] T ,u Ω =[u z ,u φ ,u θ ,u ψ ] T ,u x1 、u y1 、u z 、u y2 、u x2 、u φ 、u θ 、u ψ They represent the control quantities of the x, y, α, β, φ, θ, and ψ subsystems respectively.

[0061] d P =[d x ,d y ] T ,d Q =[d α ,d β ] T ,d Ω =[d z ,d φ ,d θ ,d ψ ] T .d x d y d z d α d β d φ d θ d ψ Represents the amount of interference received by each subsystem.

[0062] In step 3, η1 represents the observed state variable of the quadrotor suspension system, and d η Represents the system disturbance and serves as the expanded state of the system, let The observed state space expression of the quadrotor suspension system is expressed as:

[0063]

[0064] in, η2 is the first-order derivative of the system state variable with respect to time, is the second-order derivative of the system state variable with respect to time, η3 here represents the system disturbance, is the third-order expansion state of the system, where is the first-order derivative of the system disturbance with respect to time. f(η) is the model variable in the system model except for the term where the control quantity is located, and g(η) is the control quantity coefficient in the system model.

[0065] Let ∈ be the observation error, and introduce the nonlinear feedback function g n (∈):

[0066]

[0067] Where n is the system order, and the system in this part is a 3rd-order system, that is, n≤3.

[0068] The nonlinear extended state observer can be designed as:

[0069]

[0070] in, are the observed values ​​of η1, η2, and η3 respectively; μ1, μ2, and μ3 are the error feedback gains of the extended state observer. g1(∈), g2(∈), and g3(∈) are g n (∈)The expression when n=1,2,3.

[0071] In step 4, according to the control objective, combined with the linear sliding mode and non-singular terminal sliding mode theory, the following non-singular fast terminal sliding mode surfaces are designed for the x-β subsystem:

[0072]

[0073] Among them: a x 、c x 、a β 、c β 、p x ,q x 、p β ,q β These are controller parameters.

[0074] e x is the deviation between the expected trajectory and the actual trajectory of the x subsystem; for e x Derivative with respect to time.

[0075] a x 、c x 、a β 、c β >0;p x ,q x 、p β ,q β are all positive odd numbers; and p x >qx , p β >q β ;

[0076] Reaching law selection:

[0077]

[0078] in: They represent the reaching laws of x and β subsystems respectively.

[0079] k x 、k β ,η x ,η β , γ x , γ β For controller parameters.

[0080] k x 、k β ,η x ,η β >0,0<γ x <1, 0<γ β <1.

[0081] u x1 、u x2 Can be designed as:

[0082]

[0083] in: is the disturbance estimated by the observer. x1 is the control law of the x subsystem, u x2 is the control law of the β subsystem, is the second-order derivative of the expected trajectory of the x subsystem with respect to time; f(x), g(x), f(β), and g(β) are model variables, and their specific expressions are shown below.

[0084]

[0085] The swing angle variable of the suspended load cannot be controlled directly, but can only be controlled indirectly through the position control signal. The position control signal can be expressed as:

[0086] u x =u x1 +u x2

[0087] Similarly, the non-singular fast terminal sliding mode surface of the y-α control subsystem is selected as:

[0088]

[0089] Where: s y 、sα are the sliding surfaces of the y and α subsystems respectively; a y 、c y 、a α 、c α 、p y ,q y 、p α ,q α is the controller parameter; e y is the deviation between the expected trajectory and the actual trajectory of the y subsystem; for e x Derivative with respect to time.

[0090] The reaching law is:

[0091]

[0092] in: They represent the reaching laws of y and α subsystems respectively.

[0093] k y 、k α ,η y ,η α , γ y , γ α For controller parameters.

[0094] k y 、k α ,η y ,η α >0,0<γ y <1, 0<γ α <1.

[0095] u y1 、u y2 Can be designed as:

[0096]

[0097] in: is the disturbance estimated by the observer; u y1 is the control law of the y subsystem; u y2 is the control law of the α subsystem; is the second-order derivative of the expected trajectory of subsystem y with respect to time. f(y), g(y), f(α), and g(α) are model variables, and their specific expressions are shown below.

[0098]

[0099] The position control signal can be expressed as:

[0100] u y =u y1 +uy2

[0101] For the design of the remaining height and attitude subsystems, taking the height z subsystem as an example, the sliding surface can be selected:

[0102]

[0103] Where: s z is the sliding surface of the designed height and attitude subsystem; a z 、c z 、p z ,q z is the controller parameter; e z is the deviation between the expected trajectory and the actual trajectory of the altitude and attitude subsystem; for e z First derivative with respect to time.

[0104] Convergence law:

[0105]

[0106] in: represents the reaching law of the z subsystem; k z ,η z , γ z For controller parameters.

[0107] u z Can be designed as:

[0108]

[0109] Where: u Ω Output of the designed altitude and attitude subsystem controller. is the second-order derivative of the desired trajectory with respect to time; f(z) and g(z) are model variables, and the specific expressions are as follows.

[0110]

[0111] The controller designs of the remaining φ, θ, and ψ subsystems of the altitude and attitude subsystem are similar to those of the altitude z subsystem. Their controller designs are:

[0112]

[0113] Among them, f(φ), f(θ), f(ψ), g(φ), g(θ), and g(ψ) are model variables, and the specific expressions are as follows.

[0114]

[0115] The present invention proposes a sliding mode control method for a quadrotor suspension system based on an extended state observer. Compared with the existing technology, the present invention shows many obvious advantages in solving the load swing problem of quadrotor suspension UAVs.

[0116] First, in terms of control strategy, this invention innovatively introduces a nonlinear extended state observer (NESO) to accurately estimate and compensate for system disturbances and uncertain parameters in real time. This design significantly enhances the robustness of the system, enabling the drone to maintain stable control performance despite uncertainties in parameters such as moment of inertia, lift coefficient, and torque coefficient, as well as external interfering forces and torques. In contrast, traditional control methods often struggle to effectively address these uncertainties, resulting in reduced system performance.

[0117] Secondly, in terms of controller design, the present invention adopts a non-singular fast terminal sliding mode controller (NFTSM). This controller not only combines non-singularity and fast convergence properties, but also effectively solves the singularity problem in traditional terminal sliding mode control by introducing nonlinear terms and fast convergence terms, and improves the system's convergence speed.

[0118] When dealing with the problem of mutual coupling between the position subsystem and the swing angle subsystem of the drone suspension system, the control scheme proposed in the present invention regards the output of the swing angle subsystem controller as a compensation of the position control signal, and linearly combines the position subsystem and the swing angle subsystem controller to obtain the system position control signal, which effectively solves the problem of high coupling between the position and swing angle systems of the quadrotor suspension system.

[0119] In summary, this invention demonstrates unique innovation and practicality in addressing the load swing problem of quadrotor suspension drones. Based on the quadrotor suspension system dynamics model established using the Lagrange-Euler equations, and by introducing advanced control technologies and strategies such as NESO and NFTSM, this invention not only improves the system's robustness and trajectory tracking performance, but also broadens its scope and application scenarios, laying a solid foundation for the further development and application of quadrotor suspension drones. BRIEF DESCRIPTION OF THE DRAWINGS

[0120] Figure 1 Schematic diagram of the structure of the quadrotor suspension system.

[0121] Figure 2 This is the structural block diagram of the control system of the quadrotor suspension system.

[0122] Figure 3 This is the corresponding relationship diagram between the virtual torque and each subsystem after the overall system is decomposed.

[0123] Figure 4Schematic diagram of tracking the desired trajectory in the x and y directions and suppressing the load swing angle.

[0124] Figure 5 Schematic diagram of the attitude control system simulation structure.

[0125] Figure 6 Schematic diagram of the limiting filtering effect of the limiting filter.

[0126] Figure 7 Schematic diagram of the expected trajectory tracking effect for the altitude and attitude subsystems.

[0127] Figure 8 Schematic diagram of the estimated effect of the extended state observer on the assumed disturbance.

[0128] Figure 9 Schematic diagram of the jitter amplitude of the controller output when different reaching laws are selected. DETAILED DESCRIPTION

[0129] A sliding mode control method for a quadrotor suspension system based on an extended state observer addresses the common problem of load swing affecting trajectory tracking performance in quadrotor suspension systems. This innovative solution, based on nonsingular fast sliding mode control using overall system decomposition, is proposed. First, an accurate mathematical model of the quadrotor suspension system is constructed using the Lagrange-Euler equations. An advanced nonlinear extended state observer is introduced to accurately estimate system disturbances in real time. Next, to more effectively manage system complexity, a virtual torque is assigned to each state variable, and the entire system is cleverly decomposed into several subsystems. This step not only simplifies control design but also improves system controllability and responsiveness. Furthermore, based on the disturbance estimates, a nonsingular fast terminal sliding mode control method is designed for each subsystem. This approach ensures that each subsystem can quickly and stably reach the desired state when faced with disturbances. Subsequently, the corresponding relationships between the virtual torques are used to derive torque commands that meet actual control requirements. This step ensures that our control strategy meets theoretical requirements and can be effectively implemented in practical applications.

[0130] A decomposition method for the overall system and a subsequent non-singular fast sliding mode control scheme for a quadrotor suspension system based on disturbance estimation implemented for each decomposed subsystem include the following steps:

[0131] S1: Using the Lagrange-Euler equation, the dynamic model equation of the quadrotor suspension system was constructed, providing a theoretical basis for subsequent analysis and control.

[0132] S2: To achieve precise control of each state variable in the quadrotor suspension system, we conducted an in-depth analysis of the established system model. We obtained virtual control torques corresponding to each state variable, and based on these torques, we decomposed the entire system into multiple subsystems to facilitate subsequent independent control design.

[0133] S3: Considering the various uncertain disturbances that may exist in the system, these disturbances are regarded as the system's extended state variables. Based on this, a nonlinear extended state observer is designed to estimate the system's disturbances in real time.

[0134] S4: Based on the disturbance estimates provided by the nonlinear extended observer, a nonsingular fast sliding mode controller is designed for each subsystem.

[0135] The present invention provides a dynamic model of a quadrotor suspension system, the structure diagram of which is shown in FIG. Figure 1 As shown, the quadrotor suspension system includes a quadrotor drone, a suspension rope used to suspend a load, and a suspended load. The modeling process of the dynamic model of the quadrotor suspension system provided by the present invention is mainly divided into two parts. The first is to establish the Lagrange equation and combine the coordinates of the center of mass of the quadrotor drone with the coordinates of the center of mass of the suspended load to obtain the system position swing angle dynamic model; the second is to obtain the relationship between the rotational torque around the axis and the attitude angular acceleration based on the Euler equation, thereby determining the system attitude dynamic model.

[0136] The present invention provides a sliding mode control system based on the established quadcopter suspension system dynamics model, such as Figure 2 As shown, the control system includes the establishment of non-singular fast sliding mode controllers for the x, y, z, α, β, φ, θ, ψ subsystems, the combination of various sliding mode controllers, and the design of nonlinear extended state observers. The corresponding relationship between each controller and each subsystem that needs to be designed is shown in the following figure: Figure 3 shown.

[0137] The specific implementation details of step S1 of the proposed control scheme are as follows:

[0138] First, the Lagrange equations were used to construct a dynamic model for the position and swing angle system. This step describes the load's position in space and its swing characteristics. Simultaneously, an attitude dynamics model was established, combining the Euler equations, to reflect the attitude changes of the quadrotor suspension system during flight.

[0139] In order to simplify the dynamic model of the system, the following assumptions are made:

[0140] (1) The quadrotor drone is symmetrical and rigid;

[0141] (2) The load is connected to the center of mass of the quadrotor drone through a rope, and the rope is massless, inelastic, and always in a tensioned state;

[0142] (3) The payload can be considered as a point mass, which is always located below the UAV;

[0143] (4) Ignore air resistance.

[0144] The variables used to build the model are shown in Table 1 below:

[0145] Table 1 Meaning of parameters used in model establishment

[0146] symbol meaning symbol meaning x,y,z Quadrotor center of mass position φ,θ,ψ Angle of rotation around the axis M Drone quality m Load mass α Swing angle projection on the yz plane β Swing angle projection on the xz plane <![CDATA[u1]]> Four-rotor lift torque <![CDATA[u φ ,in θ ,in ψ ]]> Torque about the axis l The distance from the center of mass of the quadrotor to the rotor axis L Length of hanging rope g Gravity <![CDATA[I x ,i y ,I z ]]> The moment of inertia of the quadrotor about its axis <![CDATA[x d ,y d ,z d ]]> Position system expected trajectory <![CDATA[φ d ,i d ,ψ d ]]> Expected trajectory of attitude system <![CDATA[e x ,And y ,And z ]]> Position system tracking error <![CDATA[e φ ,And θ ,And ψ ]]> Attitude system tracking error <![CDATA[d x ,d y ,d z ]]> Position system disturbance <![CDATA[d φ ,d θ ,d ψ ]]> Attitude system disturbance

[0147] First, based on the above assumptions, the position swing angle system dynamics model is established with the inertial coordinate system as the benchmark. The center of mass position of the quadrotor drone can be expressed as (x, y, z), and the load position coordinates can be expressed as (x p ,y p ,z p ), we can get the relationship between the coordinates of the center of mass of the payload and the coordinates of the center of mass of the quadrotor drone:

[0148]

[0149] The dynamic model of the position swing angle system established using the Lagrange equation is as follows:

[0150]

[0151] Among them, q=[x, y, z, α, β, φ, θ, ψ] T , u is the driving force of the system in the position swing subsystem, here it is the lift torque u1 of the quadrotor suspension system.

[0152]

[0153] G(q)=[00g sin(α)g cos(α)sin(β)g] T ;

[0154] b=[b1 b2 b300] T ;

[0155] The internal parameters of the M(q) matrix are defined as follows:

[0156]

[0157] The internal parameters of the matrix are defined as follows:

[0158]

[0159] The internal parameters of the b matrix are defined as follows:

[0160]

[0161] The attitude system dynamics model established using the Euler equation is as follows:

[0162]

[0163] The four-rotor suspension system adopts a cross structure. The four rotor speeds ω1, ω2, ω3, ω4 are related to the system lift torque u1 and the torque around the three axes u φ ,u θ ,u ψ The corresponding relationship is:

[0164]

[0165] The specific implementation details of step S2 of the proposed control scheme are as follows:

[0166] First, assume that there is a virtual moment u x ,u y ,u z ,

[0167]

[0168] The corresponding relationship between the position signal and the expected value of the attitude angle is solved:

[0169]

[0170] The position and swing angle dynamics model of the quadrotor suspension system can be expressed as:

[0171]

[0172] The present invention will x As the main control quantity of the β subsystem, u y As the main control quantity of the α subsystem, u x Decompose to obtain the control quantity u of the x subsystem x1 and the control quantity u of the β subsystem x2 ,u y Decomposed into the control quantity u of the y subsystem y1 and the control quantity u of the α subsystem y2 , thus we get u x 、u y The expression:

[0173] u x =u x1 +u x2

[0174] uy =u y1 +u y2 ;

[0175] In the proposed control scheme, the specific details of step S3 are as follows:

[0176] η1 represents the observed state variable of the quadrotor suspension system, and d η Represents the system disturbance and serves as the expanded state of the system, let The observed state space expression of the quadrotor suspension system is expressed as:

[0177]

[0178] in, η2 is the first-order derivative of the system state variable with respect to time, is the second-order derivative of the system state variable with respect to time, η3 here represents the system disturbance, is the third-order expansion state of the system, where is the first-order derivative of the system disturbance with respect to time. f(η) is the model variable in the system model except for the term where the control quantity is located, and g(η) is the control quantity coefficient in the system model.

[0179] Let ∈ be the observation error, and introduce the nonlinear feedback function g n (∈):

[0180]

[0181] Where n is the system order, and the system in this part is a 3rd-order system, that is, n≤3.

[0182] The nonlinear extended state observer can be designed as:

[0183]

[0184] in, are the observed values ​​of η1, η2, and η3 respectively; μ1, μ2, and μ3 are the error feedback gains of the extended state observer. g1(∈), g2(∈), and g3(∈) are g n (∈)The expression when n=1,2,3.

[0185] In the proposed control scheme, the specific details of step S4 are as follows:

[0186] According to the control objective, combined with the linear sliding mode and non-singular terminal sliding mode theory, the following non-singular fast terminal sliding mode surfaces can be designed for the x-β subsystem:

[0187]

[0188] Among them: ax 、c x 、a β 、c β >0,p x ,q x 、p β ,q β are all positive odd numbers, and p x >q x , p β >q β .

[0189] Reaching law selection:

[0190]

[0191] Where: k x 、k β ,η x ,η β >0,0<γ x <1, 0<γ β <1.

[0192] u x1 、u x2 Can be designed as:

[0193]

[0194] Where: d x is the disturbance estimated by the observer.

[0195] Similarly, the non-singular fast terminal sliding mode surface of the y-α control subsystem is selected as:

[0196]

[0197] The reaching law is:

[0198]

[0199] u y1 、u y2 Can be designed as:

[0200]

[0201] The design of the height and attitude subsystem controller, taking the height z subsystem as an example, can select the sliding surface:

[0202]

[0203] Where: s z is the sliding surface of the designed height and attitude subsystem; a z 、c z 、p z,q z is the controller parameter; e z is the deviation between the expected trajectory and the actual trajectory of the altitude and attitude subsystem; for e z First derivative with respect to time.

[0204] Convergence law:

[0205]

[0206] in: represents the reaching law of the z subsystem; k z ,η z , γ z For controller parameters.

[0207] u z Can be designed as:

[0208]

[0209] Where: u Ω Output of the designed altitude and attitude subsystem controller. is the second-order derivative of the given height desired trajectory with respect to time, f(z) and g(z) are model variables, and the specific expressions are as follows.

[0210]

[0211] The controller designs of the remaining φ, θ, and ψ subsystems of the altitude and attitude subsystem are similar to those of the altitude z subsystem. Their controller designs are:

[0212]

[0213] Among them, f(φ), f(θ), f(ψ), g(φ), g(θ), and g(ψ) are model variables, and the specific expressions are as follows.

[0214]

[0215] The effectiveness of the proposed control scheme is verified using simulation results. During the simulation experiment, the system's initial states were all set to zero, and the simulation duration was set to 20 seconds. To comprehensively evaluate the proposed algorithm's performance in trajectory tracking and its effect on swing angle suppression, the α and β curves under its control were compared with those of a control system without swing angle suppression (i.e., without the α and β subsystem controllers mentioned in this invention).

[0216] The parameters of the system model, each controller and the extended state observer are shown in Tables 2 to 5 below.

[0217] Table 2 Quadrotor suspension system model parameters

[0218]

[0219] Table 3 Parameters of x-β, y-α, z subsystem controllers

[0220]

[0221] Table 4 φ, θ, ψ subsystem controller parameters

[0222]

[0223] Table 5 Parameters of the third-order extended state observer

[0224]

[0225] In order to further verify whether the algorithm can accurately track the expected position and effectively suppress the swing angle during the system motion, a set of expected trajectories is designed: d =5,y d =sin(t)+5,z d =10+t, and set the desired yaw angle to ψ d =π / 6. In addition, assume that each subsystem of the system is subject to the following disturbance: d x =0.2sin(t),d y =0.2cos(t),d z =0.5sin(t),d φ =cos(t),d θ =sin(t),d ψ =1.

[0226] Figure 4 Detailed simulation results in the x and y directions provide a clear demonstration of the algorithm's performance. When the system performs intermittent point-to-point motion, the algorithm quickly and accurately guides the system to the target position while effectively and efficiently suppressing the load's angular swing. During reciprocating motion, the swing suppressor also demonstrates excellent control capabilities, strictly limiting the load's swing to a pre-set safety range, effectively avoiding potential safety hazards.

[0227] In the actual simulation experiment of the attitude system, it can be found that the result φ obtained by the attitude solution d and θ d The rate of change is quite rapid. When using this data to design a sliding mode controller, it is found that the required high-order derivatives and At some point, it approaches infinity, which brings difficulties to the design of attitude control law. dand θ d In order to ensure that this processing does not interfere with the normal operation of the position swing angle controller, a filtering processing step is added to the simulation process.

[0228] Specifically, a fourth-order Butterworth low-pass filter is used in the simulation structure, and the edge filter frequency is set to 30 rad / s to effectively filter out high-frequency noise signals, such as Figure 5 In addition, in order to further enhance flight safety, a limiter is added to the output of the attitude solution, and the edge amplitude is set to 0.3, thereby ensuring that the desired attitude angle of the drone is limited to within 20°, effectively preventing the safety risks that may be caused by excessive attitude angles. The effect of the limiting filter is shown in the figure below. Figure 6 shown.

[0229] After the attitude angle expected signal is processed by limiting filtering, Figure 7 It can be clearly seen from the (a), (b) and (c) sub-graphs in Figure 1 that the designed controller can track the expected value of the attitude angle after filtering. In addition, Figure 7 Subfigure (d) in Figure 3 further demonstrates the excellent performance of the height controller in trajectory tracking.

[0230] Nonlinear extended state observers are designed for the six key subsystems: positions x, y, z, and attitude angles φ, θ, ψ. These observers can estimate the amount of disturbance experienced by each subsystem with considerable accuracy. Figure 8 The estimation results of these perturbations are presented intuitively.

[0231] While the controller achieves the control target, the system chattering is significantly suppressed thanks to the specially designed reaching law. Figure 9 As shown, compared with the traditional reaching law s=-ks-ηsign(s), the nonlinear reaching law adopted in the present invention makes u x and u y The trend of change is more stable.

[0232] The present invention involves a method for decomposing the overall system and designing a nonsingular fast sliding mode control strategy for a quadrotor suspension system based on disturbance estimation for each of the decomposed subsystems. This control scheme aims to enable the quadrotor suspension system to accurately track the desired target while effectively and rapidly suppressing payload oscillation.

Claims

1. A sliding mode control method for a quadrotor suspension system based on an extended state observer, characterized in that The following steps are involved: Step 1: Construct a dynamic model of the quadrotor suspension system; Step 2: Analyze the dynamic model of the quadrotor suspension system established in step 1 to obtain the virtual control torque corresponding to each state variable of the quadrotor suspension system. Based on this, decompose the entire quadrotor suspension system into multiple subsystems. Step 3: Consider the various uncertain disturbances in the quadrotor suspension system as the system's extended state variables. Based on this, a nonlinear extended state observer is designed to estimate the system's disturbances in real time. Step 4: Design a nonsingular fast sliding mode controller for each subsystem based on the disturbance estimate provided by the nonlinear extended state observer; In step 3, Let ∈ be the observation error, and introduce the nonlinear feedback function g n (∈): Where n is the system order; The nonlinear extended state observer can be designed as: in, are the observed values ​​of η1, η2, and η3 respectively; μ1, μ2, and μ3 are the error feedback gains of the extended state observer; g1(∈), g2(∈), and g3(∈) are the g n (∈) expression when n=1,2,3; In the step 4, For the x-β subsystem, the following non-singular fast terminal sliding mode surfaces are designed: Among them: a x 、c x 、a β 、c β 、p x ,q x 、p β ,q β All are controller parameters; e x is the deviation between the expected trajectory and the actual trajectory of the x subsystem; for e x derivatives with respect to time; a x 、c x 、a β 、c β >0;p x ,q x 、p β ,q β are all positive odd numbers; and p x >q x , p β >q β ; u x1 、u x2 Can be designed as: in: is the disturbance estimated by the observer; u x1 is the control law of the x subsystem, u x2 is the control law of the β subsystem, is the second-order derivative of the expected trajectory of the x subsystem with respect to time; f(x), g(x), f(β), g(β) are model variables; Similarly, the non-singular fast terminal sliding mode surface of the y-α control subsystem is selected as: Where: s y 、s α are the sliding surfaces of the y and α subsystems respectively; a y 、c y 、a α 、c α 、p y ,q y 、p α ,q α is the controller parameter; e y is the deviation between the expected trajectory and the actual trajectory of the y subsystem; for e x derivatives with respect to time; u y1 、u y2 Can be designed as: in: is the disturbance estimated by the observer; u y1 is the control law of the y subsystem; u y2 is the control law of the α subsystem; is the second-order derivative of the expected trajectory of the y subsystem with respect to time; f(y), g(y), f(α), g(α) are model variables; For the height z subsystem, select the sliding surface: Where: s z is the sliding surface of the designed height and attitude subsystem; a z 、c z 、p z ,q z is the controller parameter; e z is the deviation between the expected trajectory and the actual trajectory of the altitude and attitude subsystem; for e z First derivative with respect to time; u z Designed to: Where: u Ω Output of the designed altitude and attitude subsystem controller; is the second-order derivative of the desired trajectory with respect to time; f(z) and g(z) are model variables.

2. The sliding mode control method for a quadrotor suspension system based on an extended state observer according to claim 1, characterized in that: In step 1, the quadrotor suspension system includes a quadrotor drone, a suspension rope used to suspend a load, and a suspended load. The modeling process of the dynamic model of the quadrotor suspension system includes: By using the Lagrange equation and combining the relationship between the coordinates of the center of mass of the quadrotor drone and the center of mass of the suspended load, the system position swing angle dynamic model is obtained to describe the position of the load in space and its swing characteristics. According to the Euler equation, the relationship between the rotational torque around the axis and the attitude angular acceleration is obtained, and then the system attitude dynamics model is determined to reflect the attitude changes of the quadrotor suspension system during flight.

3. The sliding mode control method for a quadrotor suspension system based on an extended state observer according to claim 2, characterized in that: Based on the inertial coordinate system, the system position swing angle dynamic model is established. It is assumed that the center of mass position of the quadrotor drone is expressed as (x, y, z) in the inertial coordinate system, and the load position is expressed as (x p ,y p ,z p ), the relative relationship between the coordinates of the center of mass of the payload and the coordinates of the center of mass of the quadrotor drone is obtained as follows: Where: L is the length of the hanging rope; α is the swing angle projection on the yz plane, and β is the swing angle projection on the xz plane; The system position swing angle dynamic model established using the Lagrange equation is as follows: Among them, q=[x, y, z, α, β, φ, θ, ψ] T , [·] T is the transpose of the vector [·], is the first-order derivative of q with respect to time, is the second-order derivative of q with respect to time, u is the driving force of the system in the position swing subsystem, here it is the lift moment u1 of the quadrotor suspension system; M(q), G(q) and b are the definitions of the model parameter matrix; The system attitude dynamics model established using the Euler equation is as follows: Where l is the wheelbase of the quadrotor drone; φ, θ, ψ are the angles of the drone’s counterclockwise rotation around the body’s coordinate axis; are the second-order derivatives of φ, θ, and ψ with respect to time; are the first-order derivatives of φ, θ, and ψ with respect to time; u φ ,u θ ,u ψ They represent the counterclockwise rotation torque of the drone around the body coordinate axis; I x ,I y ,I z They represent the moment of inertia of the UAV rotating counterclockwise around the body coordinate axis; d φ ,d θ ,d ψ is the uncertainty disturbance of the system; The four-rotor suspension system adopts a cross structure. The four rotor speeds ω1, ω2, ω3, ω4 are related to the system lift torque u1 and the torque around the three axes u φ ,u θ ,u ψ The corresponding relationship is: Among them, c T is the lift coefficient, c M is the counter torque coefficient.

4. The sliding mode control method for a quadrotor suspension system based on an extended state observer according to claim 1, characterized in that: In step 2, first, set the virtual control torque u x ,u y ,u z : Where M represents the mass of the drone and m represents the mass of the payload; Then the virtual control torque u can be obtained x ,u y ,u z With the system lift torque u1 and the desired attitude angle φ d ,θ d The relationship between: The position and swing angle dynamics model of the quadrotor suspension system can be expressed as: in, Represent the second-order derivatives of x, y, and z with respect to time, are the second and first order derivatives of α and β with respect to time, respectively, d x ,d y ,d z ,d α ,d β represents the uncertainty disturbance of the system, g represents the acceleration of gravity; will u x As the main control quantity of the β subsystem, u y As the main control quantity of the α subsystem, u x Decompose to obtain the control quantity u of the x subsystem x1 and the control quantity u of the β subsystem x2 ,u y Decomposed into the control quantity u of the y subsystem y1 and the control quantity u of the α subsystem y2 , thus we get u x 、u y The expression: Let P = [x, y] T , Q=[α,β] T ,Ω=[z,φ,θ,ψ] T , the entire quadrotor suspension system can be redefined as: in, is the second-order derivative of P, Q, and Ω with respect to time; P, Q, and Ω represent the state variable vectors of the translation subsystem, the swing subsystem, and the height and attitude subsystem, respectively; f(P), f(Q), and f(Ω) are the model variables other than the control quantity in the translation subsystem model, the swing subsystem model, and the height and attitude subsystem model, respectively; g(P), g(Q), and g(Ω) are the control quantity coefficients of the translation subsystem model, the swing subsystem model, and the height and attitude subsystem model, respectively; u P 、u Q 、u Ω They are the control quantities of translation subsystem, swing subsystem, height and attitude subsystem respectively.

5. The sliding mode control method for a quadrotor suspension system based on an extended state observer according to claim 1, characterized in that: In step 3, η1 represents the observed state variable of the quadrotor suspension system, and d η Represents the system disturbance and serves as the expanded state of the system, let The observed state space expression of the quadrotor suspension system is expressed as: in, η2 is the first-order derivative of the system state variable with respect to time, is the second-order derivative of the system state variable with respect to time, η3 here represents the system disturbance, is the third-order expansion state of the system, which is the first-order derivative of the system disturbance with respect to time; f(η) is the model variable in the system model except the term where the control quantity is located, and g(η) is the control quantity coefficient in the system model.

6. The sliding mode control method for a quadrotor suspension system based on an extended state observer according to claim 1, characterized in that: In step 4, the reaching law of the x-β subsystem is selected as: in: They represent the reaching laws of x and β subsystems respectively; k x 、k β ,η x ,η β , γ x , γ β is the controller parameter; k x ,k β ,or x ,or β >0,0<γ x <1,0<γ β <1; The specific expressions of model variables f(x), g(x), f(β), and g(β) are as follows; The swing angle variable of the suspended load cannot be controlled directly, but can only be controlled indirectly through the position control signal. The position control signal is expressed as: in x =in x1 +in x2 ; The reaching law of the y-α control subsystem is: in: They represent the reaching laws of y and α subsystems respectively; k y 、k α ,η y ,η α , γ y , γ α is the controller parameter; k y ,k α ,or y ,or α >0,0<γ y <1,0<γ α <1; The specific expressions of model variables f(y), g(y), f(α), and g(α) are as follows; The position control signal is expressed as: in y =in y1 +in y2 ; For the design of the remaining height and attitude subsystems, for the height z subsystem, the reaching law of the height z subsystem is: in: represents the reaching law of the z subsystem; k z ,η z , γ z is the controller parameter; The specific expressions of model variables f(z) and g(z) are as follows; The controller designs of the remaining φ, θ, and ψ subsystems of the altitude and attitude subsystem are similar to those of the altitude z subsystem. Their controller designs are: Among them, f(φ), f(θ), f(ψ), g(φ), g(θ), and g(ψ) are model variables, and their specific expressions are as follows;

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