Self-adaptive fault-tolerant control design method for quad-rotor unmanned aerial vehicle with drift measurement function

Abstract

The invention discloses a self-adaptive fault-tolerant control design method for a quad-rotor unmanned aerial vehicle with measurement drift in the field of low-altitude economy, solves the problem that the system state cannot be directly measured due to the measurement drift of a quad-rotor unmanned aerial vehicle system, and improves the fault-tolerant performance of the system. Due to the fact that the system state cannot be directly measured due to measurement drift, an error signal is established by introducing a new auxiliary variable, and the influence of a drift item is compensated; in order to solve the problems of uncertainty and unknown disturbance of parameters of the quad-rotor unmanned aerial vehicle, all unknown parameters and interference of a system are effectively estimated through a radial basis function neural network, the uncertainty and unknown disturbance of the unknown parameters are strictly proved, and the designed controller enables the system to be stable; according to the method, a backstepping control method is adopted, an adaptive law and a controller of the quad-rotor unmanned aerial vehicle under unknown measurement drift are obtained, simulation shows that the method has a good control effect, it is guaranteed that tracking errors are bounded, and good tracking performance is obtained. The four-rotor unmanned aerial vehicle system control method has good implementability, can generate obvious economic benefits, and improves the practicability of the four-rotor unmanned aerial vehicle system control algorithm in the aspect of engineering practical application.

Description

Technical Field

[0001] This invention relates to the field of low-altitude economy, and in particular to an adaptive fault-tolerant control design method for a quadcopter unmanned aerial vehicle with measurement drift. Background Technology

[0002] In today's era of rapid technological advancement, drone technology is undergoing rapid development. Drones, or unmanned aerial vehicles (UAVs), are aircraft that can autonomously perform flight missions without direct human intervention. Initially, they were primarily used in the military field, including target reconnaissance, battlefield surveillance, and unmanned combat missions. In recent years, with the rapid rise of civilian drone companies like DJI, drones have gained widespread attention in the civilian market. Their applications have expanded to various industries such as film and television aerial photography, logistics and transportation, forest fire prevention, disaster relief, inspection and monitoring, and geographic surveying, gradually becoming an important technological force driving industry innovation and efficiency improvement. From a structural perspective, drones can be mainly divided into two categories: fixed-wing and rotary-wing. Fixed-wing drones have been developed earlier and are relatively mature in technology, possessing advantages such as high-speed flight and good stability. In contrast, multi-rotor drones—especially quadcopters—have received widespread attention from academia and industry due to their compact structure, vertical takeoff and landing capabilities, low cost, ease of manufacturing, maneuverability, and strong environmental adaptability. Related control technologies have also made significant progress, and they have occupied an important position in many application scenarios.

[0003] The working principle of a quadcopter drone primarily relies on the rotational speed adjustment of its four rotors. By changing the rotational speed of different rotors, different lift and torque can be generated, thereby achieving attitude and position control of the aircraft. However, its practical application still faces challenges in many aspects, including safety and stability, especially in the sensor measurement stage. As a key component of the drone control system for sensing external states, the output accuracy of sensors is easily affected by the measuring tools, technical level, and environmental noise, leading to deviations in the measurement of system states (such as angle, distance, temperature, etc.). Specifically, the relationship between sensor input and output often exhibits time-varying, nonlinear, or even completely unknown characteristics, causing unknown drift in the power of the output function. For example, the voltage signal output by an infrared distance sensor in actual distance detection is not only nonlinear but also accompanied by significant noise, and its output often deviates from the true physical value, resulting in inaccurate readings for actual distance measurements. The sensor only outputs a constant. Approximately 0.8 Similar phenomena have also appeared in measurement systems of automobiles and aircraft, indicating that measurement drift is a common problem in many industrial control systems. This phenomenon places higher demands on the stability and safety control of UAVs. In recent years, scholars such as Meng and Ma have conducted research on the stability control of such systems by constructing novel integrators, providing theoretical support for the control design of quadrotor UAVs in the presence of measurement drift. The aforementioned literature has certain guiding significance for our research on the stability control of quadrotor UAV systems under such conditions.

[0004] Since quadcopter UAVs are inherently multi-input multi-output, underactuated nonlinear systems, traditional control methods struggle to achieve bounded tracking errors in the presence of measurement drift. Therefore, developing an adaptive fault-tolerant control design method capable of compensating for measurement drift is of significant practical importance. This invention proposes an adaptive fault-tolerant control scheme that effectively addresses the uncertainties caused by measurement drift by real-time estimation and adjustment of the quadcopter UAV's state parameters, thereby ensuring bounded tracking errors in complex environments. Simulation experiments demonstrate that, even with measurement drift, the quadcopter UAV using this controller can still accurately and stably track the predetermined trajectory, verifying the effectiveness and engineering practicality of the proposed control strategy. This research provides strong technical support for the reliable application of high-order, high-dynamic-performance UAV systems in real-world environments. Summary of the Invention

[0005] To address the shortcomings of existing technologies, improve the accuracy of sensor measurements, and consider the relationship between sensor input and output, this invention provides an adaptive fault-tolerant control design method for a quadcopter UAV with measurement drift.

[0006] The objective of this invention is achieved as follows: an adaptive fault-tolerant control design method for a quadcopter unmanned aerial vehicle with drift measurement, comprising the following steps:

[0007] Step 1) Extend the classic quadcopter UAV dynamic equations into a second-order nonlinear system model of a UAV system with measurement drift, taking the position and attitude of the UAV's center of mass in the inertial coordinate system as its state;

[0008] Step 2) Introduce new auxiliary variables to establish error signals for the displacement subsystem and attitude subsystem to compensate for the influence of the drift term;

[0009] Step 3) Utilize the powerful approximation capability of radial basis neural networks to process unknown nonlinear terms, thereby reducing the computational load and increasing the design universality;

[0010] Step 4) Use the backstepping method to design the final controller and adaptive law for the displacement subsystem and attitude subsystem.

[0011] Furthermore, the classic quadcopter UAV dynamic equations described in step 1) are as follows:

[0012]

[0013] Where x, y, and z are the positions of the UAV's center of mass in the inertial coordinate system; These are pitch angle, roll angle, and yaw angle, respectively. g is the acceleration due to gravity; This is the aerodynamic damping coefficient; It is the moment of inertia; Indicates the rotor's inertia; The control input is generated by four rotors; The total remaining rotor angles, considered as bounded disturbances, are related as follows:

[0014]

[0015]

[0016]

[0017]

[0018] in, It is the distance between the center of mass and the rotor shaft. It is the drag coefficient. It is the reverse torque coefficient;

[0019] For ease of controller design, the following definition is provided:

[0020]

[0021]

[0022]

[0023]

[0024]

[0025]

[0026]

[0027]

[0028]

[0029] in These represent the virtual control inputs for the lateral, longitudinal, and height channels, respectively; considering the influence of external disturbances on position and attitude, such as... , The position of the UAV's center of mass in the inertial coordinate system and posture The second-order nonlinear system model of the UAV system with measurement drift is as follows:

[0030]

[0031] in For measurement output; For measuring drift, is a continuously differentiable function that satisfies ,in It is a positive number; .

[0032] Furthermore, the specific steps for introducing new auxiliary variables to establish error signals for the displacement and attitude subsystems in step 2) are as follows:

[0033] Define the error vector of the displacement subsystem Define the error vector of the rotating subsystem. ,in , For the design of the virtual controller in subsequent projects; These are the desired position and desired orientation, respectively, and are column vectors with a row of three rows and one column:

[0034] The error vector of the displacement subsystem is obtained by indirectly processing it using virtual parameters:

[0035]

[0036] The attitude subsystem error vector is obtained by indirectly processing it using virtual parameters.

[0037]

[0038] Furthermore, the neural network described in step 3) is

[0039]

[0040] in =1, 2, 3, 4 It is an unknown nonlinear term. It is the input to the radial basis function neural network; It is the target weight matrix of the output layer; It is the total number of neurons in the hidden layer; This is the corresponding reconstruction error.

[0041] Furthermore, the final controller and adaptive law design schemes for the displacement subsystem and rotation subsystem described in step 4) are as follows:

[0042] 5-1) Displacement Subsystem

[0043] 5-1-1) Selecting Lyapunov functions for the displacement subsystem:

[0044]

[0045] in ,and yes The estimation error; It is an adjustable parameter. The virtual controller and adaptive law are then designed in the following form:

[0046]

[0047]

[0048] in, , It is an adjustable parameter. Then, by using dummy parameters to indirectly process the function... The derivative becomes

[0049]

[0050] Virtual control law Substituting into the above formula, we get

[0051]

[0052] According to the drift formula It can be known that The transformation yields:

[0053]

[0054]

[0055] Further construct a neural network to estimate the unknown nonlinear term:

[0056]

[0057] in Substitute the neural network and adaptive law into the function. The derivative can be obtained as follows:

[0058]

[0059] in

[0060] ;

[0061] 5-1-2) Selecting Lyapunov functions for the displacement subsystem:

[0062]

[0063] in ,and yes The estimation error; It is an adjustable parameter.

[0064] The controller for the displacement subsystem is designed as follows:

[0065]

[0066] in It is an adjustable parameter. The adaptive rate design of the displacement subsystem is as follows:

[0067]

[0068] in It is an adjustable parameter. Then, by using dummy parameters to indirectly process the function... The derivative becomes

[0069]

[0070] controller Substituting, we get:

[0071]

[0072] According to the drift formula It can be known that ,get

[0073]

[0074]

[0075] Further construct a neural network to estimate the unknown nonlinear term:

[0076]

[0077] in Substitute the constructed neural network and adaptive law into the function. The derivative can be obtained as follows:

[0078]

[0079] in ;

[0080] 5-2) Rotating Subsystem

[0081] 5-2-1) Choosing a Lyapunov function for the rotating subsystem:

[0082]

[0083] in ,and yes The estimation error; It is an adjustable parameter. Then design the virtual controller and adaptive law as

[0084]

[0085]

[0086] in, , It is an adjustable parameter. Then, by using dummy parameters to indirectly process the function... The derivative becomes

[0087]

[0088] Virtual control law Substituting into the above formula, we get

[0089]

[0090] According to the drift formula , can be obtained , The transformation yields:

[0091]

[0092]

[0093] Further construct a neural network to estimate the unknown nonlinear term:

[0094]

[0095] in Substitute the neural network and adaptive law into the function. The derivative can be obtained as follows:

[0096]

[0097] in ;

[0098] 5-2-2) Choosing Lyapunov functions for the rotating subsystem:

[0099]

[0100] in ,and yes The estimation error; then the function Differentiating, we get:

[0101]

[0102] controller Substituting, we get:

[0103]

[0104] According to the drift formula It can be known that Substitute get

[0105]

[0106]

[0107] Further construct a neural network to estimate the unknown nonlinear term:

[0108]

[0109] in Substitute the constructed neural network and adaptive law into the function. The derivative can be obtained as follows:

[0110]

[0111] in .

[0112] Compared with the prior art, the beneficial effects of the present invention are as follows:

[0113] (1) This invention is aimed at quadcopter unmanned aerial vehicle (UAV) systems. By introducing new auxiliary variables to establish error signals and compensate for the influence of drift terms, it solves the problem that the system state cannot be directly measured due to measurement drift in UAV systems and improves the fault tolerance performance of the system.

[0114] (2) In order to deal with the uncertainty and unknown disturbance of the parameters of the quadcopter UAV, the present invention effectively estimates all unknown parameters and disturbances of the system through radial basis neural network. Compared with the traditional method, which requires updating the weight vector of the entire neural network, the method proposed in this specification only needs to update the adaptive law, which is a scalar. This reduces the number of parameters and avoids numerical drift or instability caused by updating a large number of parameters.

[0115] (3) This invention employs a backstepping control method to obtain the adaptive law and controller for a quadcopter UAV with unknown measurement drift. Simulation results show that the controller has good control performance, ensures bounded tracking error, and achieves good tracking performance. This method has good feasibility, can generate significant economic benefits, and improves the practicality of quadcopter UAV system control algorithms in engineering applications. Attached Figure Description

[0116] To more clearly illustrate the technical solutions in the embodiments of the present invention or the prior art, the drawings used in the description of the embodiments or the prior art will be briefly introduced below. Obviously, the drawings described below are only embodiments of the present invention. For those skilled in the art, other drawings can be obtained based on the provided drawings without creative effort.

[0117] Figure 1 This is a flowchart of one embodiment of the present invention.

[0118] Figure 2 This is a signal curve diagram showing the actual position and desired position of a quadcopter drone according to an embodiment of the present invention.

[0119] Figure 3 This is a signal curve diagram showing the actual attitude and desired attitude of a quadcopter drone according to an embodiment of the present invention.

[0120] Figure 4 This is a signal curve diagram of the position error of a quadcopter drone according to an embodiment of the present invention.

[0121] Figure 5 This is a signal curve diagram of the attitude error of a quadcopter drone according to an embodiment of the present invention. Detailed Implementation

[0122] The technical solutions of the embodiments of the present invention will be clearly and completely described below with reference to the accompanying drawings. Obviously, the described embodiments are only some embodiments of the present invention, and not all embodiments. Based on the embodiments of the present invention, all other embodiments obtained by those skilled in the art without creative effort are within the scope of protection of the present invention.

[0123] Example 1

[0124] An adaptive fault-tolerant control design method for a quadrotor UAV with measurement drift is presented, with the following specific steps:

[0125] Step 1) Extend the classic quadcopter UAV dynamic equations into a second-order nonlinear system model of a UAV system with measurement drift, taking the position and attitude of the UAV's center of mass in the inertial coordinate system as its state. The specific operation is as follows:

[0126] The classic quadcopter drone dynamics equation is:

[0127]

[0128] Where x, y, and z are the positions of the UAV's center of mass in the inertial coordinate system; These are pitch angle, roll angle, and yaw angle, respectively. g is the acceleration due to gravity; This is the aerodynamic damping coefficient; It is the moment of inertia; Indicates the rotor's inertia; The control input is generated by four rotors; The total remaining rotor angles, considered as bounded disturbances, are related as follows:

[0129]

[0130]

[0131]

[0132]

[0133] in, It is the distance between the center of mass and the rotor shaft. It is the drag coefficient. It is the reverse torque coefficient, taking into account the influence of external disturbances on position and attitude, such as... , Rewrite the equation as

[0134]

[0135] in , For measurement output; For measuring drift, is a continuously differentiable function that satisfies ,in It is a positive number; These represent the virtual control inputs for the horizontal, vertical, and height channels, respectively. , ;

[0136]

[0137]

[0138]

[0139]

[0140]

[0141] in,

[0142]

[0143] .

[0144] Due to the underactuated nature of the outputs, it is impossible to track all six outputs. Therefore, this embodiment selects tracking as a reasonable control objective. and yaw angle And maintaining consistency with the other two angles, at this point, , , , , , , , Both are bounded.

[0145] Step 2) Introduce new auxiliary variables to establish error signals for the displacement and rotation subsystems to compensate for the influence of the drift term. The specific operation is as follows:

[0146] Define the error vector of the displacement subsystem Define the error vector of the rotating subsystem. ,in , For the design of the virtual controller in subsequent projects; These are the desired position and desired orientation, respectively, and are column vectors with a row of three rows and one column:

[0147] The error vector of the displacement subsystem is obtained by indirectly processing it using virtual parameters:

[0148]

[0149] The attitude subsystem error vector is obtained by indirectly processing it using virtual parameters.

[0150]

[0151] Step 3) Utilize the powerful approximation capability of radial basis function neural networks to process the unknown nonlinear terms. The specific operation is as follows:

[0152] Neural networks are ,in =1, 2, 3, 4 It is an unknown nonlinear term. It is the input to the radial basis function neural network; It is the target weight matrix of the output layer; It is the total number of neurons in the hidden layer; This is the corresponding reconstruction error.

[0153] Step 4) Design the final controller and adaptive law for the displacement subsystem and rotation subsystem using the backstepping method. The specific operation is as follows:

[0154] 4-1) Design controller and adaptive law for displacement subsystem

[0155] 4-1-1) Step 1: Select a Lyapunov function for the displacement subsystem:

[0156]

[0157] in ,and yes The estimation error; It is an adjustable parameter. The virtual controller and adaptive law are then designed in the following form:

[0158]

[0159]

[0160] in, , It is an adjustable parameter. In the field of adaptive control, variables It is usually used for estimation ; , , can be obtained ; then By performing derivative processing, we can obtain:

[0161]

[0162] Substituting the virtual control law into the above equation, we get:

[0163]

[0164] According to the drift formula It can be known The transformation yields:

[0165]

[0166]

[0167] Further construct a neural network to estimate the unknown nonlinear term:

[0168]

[0169] in Substitute the neural network and adaptive law into the function. The derivative can be obtained as follows:

[0170]

[0171] in ;

[0172] 4-1-2) Step 2: Select Lyapunov functions for the displacement subsystem:

[0173]

[0174] in ,and yes The estimation error; It is an adjustable parameter. .

[0175] The controller and adaptive rate of the displacement subsystem are designed as follows:

[0176]

[0177]

[0178] in It is an adjustable parameter. Then the function The derivative becomes:

[0179]

[0180] controller Substituting, we get:

[0181]

[0182] According to the drift formula It can be known ,get

[0183]

[0184]

[0185] Further construct a neural network to estimate the unknown nonlinear term:

[0186]

[0187] in Substitute the constructed neural network and adaptive law into the function. The derivative can be obtained as follows:

[0188]

[0189] in .

[0190] based on and Calculations show that:

[0191]

[0192] The required roll and pitch references can be generated based on the virtual position controller. and Quadcopter drones have six degrees of freedom, namely ,in Based on the above control law design, the required yaw trajectory is usually given in advance. As an additional reference, therefore, and It can be interpreted as:

[0193]

[0194]

[0195] 4-2) Design controllers and adaptive laws for rotating subsystems

[0196] 4-2-1) Step 1: Select a Lyapunov function for the rotating subsystem:

[0197]

[0198] in ,and yes The estimation error; It is an adjustable parameter. .

[0199] The controller and adaptive rate of the rotating subsystem are designed as follows:

[0200]

[0201]

[0202] in It is an adjustable parameter. Then, for the function... Differentiate:

[0203]

[0204] controller Substituting, we get:

[0205]

[0206] According to the drift formula It can be known that , ,get

[0207]

[0208]

[0209] Further construct a neural network to estimate the unknown nonlinear term:

[0210]

[0211] in Substitute the constructed neural network and adaptive law into the function. The derivative can be obtained as follows:

[0212]

[0213] in .

[0214] 4-2-2) Step 1: Select a Lyapunov function for the rotating subsystem:

[0215]

[0216] in ,and yes The estimation error; It is an adjustable parameter. .

[0217] The controller and adaptive rate of the rotating subsystem are designed as follows:

[0218]

[0219]

[0220] in It is an adjustable parameter. Then the function The derivative becomes:

[0221]

[0222] controller Substituting, we get:

[0223]

[0224] According to the drift formula It can be known that ,get

[0225]

[0226]

[0227] Further construct a neural network to estimate the unknown nonlinear term:

[0228]

[0229] in Substitute the constructed neural network and adaptive law into the function. The derivative can be obtained as follows:

[0230]

[0231] in .

[0232] Example 2

[0233] An adaptive fault-tolerant control design method for a quadrotor UAV with measurement drift is simulated, and the steps are as follows:

[0234] Step 1) Select the parameters of the quadcopter drone, set the initial state of the drone and the desired trajectory of the drone.

[0235] Step 2) Establish intermediate virtual control signals for the displacement subsystem of the quadcopter UAV Build control laws Update parameter estimates and set parameter information, including .

[0236] Step 3) Establish intermediate virtual control signals for the attitude subsystem of the quadcopter UAV Build control laws Update parameter estimates and set parameter information, including .

[0237] Step 4) Build the corresponding system in Matlab / Simulink and set the relevant parameters and initial system values ​​to obtain the final simulation results.

[0238] Example 3

[0239] The specific operation process of an adaptive fault-tolerant control design method for a quadcopter UAV with drift measurement is as follows:

[0240] 1. To verify the performance of the proposed method, the position and attitude tracking of a quadcopter UAV was simulated in MATLAB / Simulink.

[0241] Table 1 lists the relevant parameters used for testing.

[0242]

[0243] Note:" "" indicates that the value has no unit; the expected yaw angle is The expected trajectory is .

[0244] 2. Establish intermediate virtual control signals for the displacement subsystem of a quadcopter UAV. Build control laws , Update parameter estimates and set parameter information. .

[0245] 3. Establish intermediate virtual control signals for the attitude subsystem of the quadcopter UAV. Establish control laws , Update parameter estimates and set parameter information. .

[0246] 4. By simulating the algorithm of this invention according to the above parameters, the following results can be obtained: Figure 2 , Figure 3 , Figure 4 , Figure 5 The diagrams show the signal curves of the actual position and desired position of the quadcopter drone, the signal curves of the actual attitude and desired attitude of the quadcopter drone, the signal curves of the position error of the quadcopter drone, and the signal curves of the attitude error of the quadcopter drone.

[0247] The above description of the embodiments is only for the purpose of helping to understand the method and core ideas of the present invention. It should be noted that those skilled in the art can make several improvements and modifications to the present invention without departing from the principles of the present invention, and these improvements and modifications also fall within the protection scope of the claims of the present invention.

Claims

1. An adaptive fault-tolerant control design method for a quadcopter unmanned aerial vehicle with drift measurement, characterized in that, Includes the following steps: Step 1) Extend the classic quadcopter UAV dynamic equations into a second-order nonlinear system model of a UAV system with measurement drift, taking the position and attitude of the UAV's center of mass in the inertial coordinate system as its state; Step 2) Introduce new auxiliary variables to establish error signals for the displacement subsystem and attitude subsystem to compensate for the influence of the drift term; Step 3) Utilize the powerful approximation capability of radial basis neural networks to process unknown nonlinear terms, thereby reducing the computational load and increasing the design universality; Step 4) Use the backstepping method to design the final controller and adaptive law for the displacement subsystem and attitude subsystem.

2. The adaptive fault-tolerant control design method for a quadcopter UAV with measurement drift as described in claim 1, characterized in that, Step 1) The classic quadcopter UAV dynamic equations are as follows: ； Where x, y, and z are the positions of the UAV's center of mass in the inertial coordinate system; These are pitch angle, roll angle, and yaw angle, respectively. g is the acceleration due to gravity; This is the aerodynamic damping coefficient; It is the moment of inertia; Indicates the rotor's inertia; The control input is generated by four rotors; The total remaining rotor angles, considered as bounded disturbances, are related as follows: ； ； ； ； in, It is the distance between the center of mass and the rotor shaft. It is the drag coefficient. It is the reverse torque coefficient; For ease of controller design, the following definition is provided: ； ； ； ； ； ； ； ； ； in These represent the virtual control inputs for the lateral, longitudinal, and height channels, respectively; considering the influence of external disturbances on position and attitude, such as... , The position of the UAV's center of mass in the inertial coordinate system and posture The second-order nonlinear system model of the UAV system with measurement drift is as follows: ； in For measurement output; For measuring drift, is a continuously differentiable function that satisfies ,in It is a positive number; .

3. The adaptive fault-tolerant control design method for a quadcopter UAV with drift measurement as described in claim 1, characterized in that, The specific steps for introducing new auxiliary variables to establish error signals for the displacement and attitude subsystems in step 2) are as follows: Define the error vector of the displacement subsystem Define the error vector of the rotating subsystem. ,in , For the design of the virtual controller in subsequent projects; These are the desired position and desired orientation, respectively, and are column vectors with a row of three rows and one column: The error vector of the displacement subsystem is obtained by indirectly processing it using virtual parameters: ； The attitude subsystem error vector is obtained by indirectly processing it using virtual parameters. 。 4. The adaptive fault-tolerant control design method for a quadcopter UAV with measurement drift as described in claim 1, characterized in that, The neural network mentioned in step 3) is ； in =1, 2, 3, 4 It is an unknown nonlinear term. It is the input to the radial basis function neural network; It is the target weight matrix of the output layer; It is the total number of neurons in the hidden layer; This is the corresponding reconstruction error.

5. The adaptive fault-tolerant control design method for a quadcopter UAV with measurement drift as described in claim 1 or 3, characterized in that, The final controller and adaptive law design schemes for the displacement subsystem and rotation subsystem mentioned in step 4) are as follows: 5-1) Displacement Subsystem 5-1-1) Selecting Lyapunov functions for the displacement subsystem: ； in ,and yes The estimation error; It is an adjustable parameter. The virtual controller and adaptive law are then designed in the following form: ； ； in, , It is an adjustable parameter. Then, by using dummy parameters to indirectly process the function... The derivative becomes ； Virtual control law Substituting into the above formula, we get ； According to the drift formula It can be known that The transformation yields: ； ； Further construct a neural network to estimate the unknown nonlinear term: ； in Substitute the neural network and adaptive law into the function. The derivative can be obtained as follows: ； in ； 5-1-2) Selecting Lyapunov functions for the displacement subsystem: ； in ,and yes The estimation error; It is an adjustable parameter. ; The controller for the displacement subsystem is designed as follows: ； in It is an adjustable parameter. The adaptive rate design of the displacement subsystem is as follows: ； in It is an adjustable parameter. Then, by using dummy parameters to indirectly process the function... The derivative becomes ； controller Substituting, we get: ； According to the drift formula It can be known that ,get ； ； Further construct a neural network to estimate the unknown nonlinear term: ； in Substitute the constructed neural network and adaptive law into the function. The derivative can be obtained as follows: ； in ; 5-2) Rotating Subsystem 5-2-1) Choosing a Lyapunov function for the rotating subsystem: ； in ,and yes The estimation error; It is an adjustable parameter. Then design the virtual controller and adaptive law as ； ； in, , It is an adjustable parameter. Then, by using dummy parameters to indirectly process the function... The derivative becomes ； Virtual control law Substituting into the above formula, we get ； According to the drift formula , can be obtained , The transformation yields: ； ； Further construct a neural network to estimate the unknown nonlinear term: ； in Substitute the neural network and adaptive law into the function. The derivative can be obtained as follows: ； in ; 5-2-2) Choosing Lyapunov functions for the rotating subsystem: ； in ,and yes The estimation error; then the function Differentiating, we get: ； controller Substituting, we get: ； According to the drift formula It can be known that Substitute get ； ； Further construct a neural network to estimate the unknown nonlinear term: ； in Substitute the constructed neural network and adaptive law into the function. The derivative can be obtained as follows: ； in .

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