An Improved Particle Swarm Optimization-Based Non-Singleton Interval Type-II Fuzzy PID Spatial Robotic Arm Tracking Control Method

Abstract

This invention relates to an improved particle swarm optimization-based non-singleton interval type-II fuzzy PID spatial manipulator tracking control method. Specifically, it relates to a PID spatial manipulator tracking control method. The purpose of this invention is to improve the accuracy and speed of existing spatial manipulator control. The process is as follows: 1. Establish a flexible joint spatial manipulator model; determine if the spatial manipulator has a fault. If not, proceed to steps 2 to 4; if so, proceed to steps 5 to 6; 2. Construct a traditional PID controller and a non-singleton interval type-II fuzzy PID controller; 3. Obtain the optimal non-singleton interval type-II fuzzy PID controller parameters, i.e., obtain the optimal non-singleton interval type-II fuzzy PID controller; 4. Control based on three pairs of flexible joint spatial manipulator models; 5. Construct a traditional PID controller, a non-singleton interval type-II fuzzy PID controller, and a signal-compensated fuzzy PID controller; 6. Control based on five pairs of flexible joint spatial manipulator models.

Description

Technical Field

[0001] This invention relates to the field of space robotic arm control, specifically to an improved particle swarm optimization non-singleton interval type II fuzzy PID space robotic arm tracking control method, and a fault-tolerant control design based on fuzzy PID signal compensation. Background Technology

[0002] The development of the space program is of great significance for safeguarding national security, promoting scientific research, improving the quality of human life, and exploring the unknown. At the same time, as space exploration deepens and space missions become increasingly complex, higher demands are placed on space exploration equipment.

[0003] During space exploration, astronauts must overcome challenges such as space radiation, microgravity, and extreme temperatures when performing extravehicular activities (EVAs). This not only reduces mission efficiency but also seriously threatens astronaut safety. With the rapid development of space technology, the use of space robots is becoming increasingly widespread. Space robots are specialized robots used to assist or replace humans in extravehicular activities and space exploration within the space environment. A space robotic arm is a special type of space robot, similar to a human arm, typically consisting of a base and multiple linkages. Space robotic arms not only assist astronauts in movement but can also perform tasks such as equipment installation, maintenance, and replacement, reducing extravehicular activity time and ensuring astronaut safety. Furthermore, space robotic arms possess powerful capture, rendezvous, and docking capabilities as well as heavy-duty transport capabilities, making them indispensable equipment for the construction, maintenance, and operation of space stations.

[0004] The controller for a space robotic arm is crucial, directly impacting its performance, precision, and ability to successfully complete various tasks. Efficient control algorithms ensure high-precision and stable control of the robotic arm, optimize its motion trajectory and force distribution, reduce energy consumption, lower task costs, and increase task success rates. Therefore, developing advanced and efficient space robotic arm controllers is imperative. Summary of the Invention

[0005] The purpose of this invention is to improve the accuracy and speed of existing space robotic arm control, and to propose an improved particle swarm optimization non-singleton interval type II fuzzy PID space robotic arm tracking control method.

[0006] The specific process of the improved non-singleton interval type-II fuzzy PID spatial robotic arm tracking control method based on particle swarm optimization is as follows:

[0007] Step 1: Establish a model of a flexible joint spatial robotic arm;

[0008] Determine if the space robotic arm is faulty. If not, proceed to steps two through four; if it is faulty, proceed to steps five through six.

[0009] Step 2: Construct a traditional PID controller and a non-singleton interval type-II fuzzy PID controller;

[0010] Step 3: Use the particle swarm optimization algorithm to optimize the parameters of the non-singleton interval type II fuzzy PID controller to obtain the optimal non-singleton interval type II fuzzy PID controller parameters, that is, obtain the optimal non-singleton interval type II fuzzy PID controller.

[0011] Step 4: Control the flexible joint spatial manipulator model established in Step 1 based on the optimal non-singleton interval type II fuzzy PID controller;

[0012] Step 5: Construct a traditional PID controller, a non-singleton interval type II fuzzy PID controller, and a signal-compensated fuzzy PID (the signal-compensated fuzzy PID is an FPID);

[0013] Step 6: Based on the traditional PID controller, non-singleton interval type II fuzzy PID controller, and signal-compensated fuzzy PID controller constructed in Step 5, control the flexible joint spatial manipulator model established in Step 1.

[0014] The beneficial effects of this invention are as follows:

[0015] This invention designs a non-singleton interval type-two fuzzy PID controller and a fault-tolerant control method for fuzzy PID signal compensation based on a dynamic model of a planar two-link flexible joint space manipulator considering faults, and optimizes its control performance using an improved particle swarm optimization algorithm. The invention designs an improved particle swarm optimization-based non-singleton interval type-two fuzzy PID space manipulator tracking control method, mainly as follows: First, a dynamic simulation model of the planar two-link flexible joint space manipulator is established, clarifying the generalized force as the input and the generalized coordinates as the output. Furthermore, the model also considers manipulator faults and introduces a fault vector. Second, an interval type-two fuzzy PID controller is designed for the space manipulator tracking control task. The interval type-two fuzzy PID controller designed in this invention uses interval type-two fuzzy logic to optimize PID controller parameters in real time, solving the problem that traditional PID parameters cannot be adjusted in real time. For the fuzzification stage of the fuzzy system, this invention adopts a non-singleton fuzzification method, implementing the non-singleton fuzzification function through simulation software programming, and applying it to the interval type-two fuzzy PID controller. Compared with the traditional singleton fuzzification method, this method can trigger more fuzzy rules and has a stronger ability to handle uncertainty. Third, this invention employs a fault-tolerant control method with interval-type II fuzzy PID signal compensation. For the error between the faulty system output and the ideal output, the fuzzy PID compensates for the input signal, thereby correcting the system output. Simulation results show that this method can effectively handle robotic arm actuator faults. Fourth, this invention uses a particle swarm optimization algorithm to optimize the fuzzy scaling factor of the interval-type II fuzzy PID controller. Compared to traditional particle swarm optimization, this invention improves its inertia coefficient, enhancing the algorithm's early-stage global search capability and later-stage local search capability. Based on simulation results and the characteristics of the controlled object with multiple inputs and outputs, this invention proposes a new fitness calculation method to simultaneously consider system accuracy and speed. Attached Figure Description

[0016] Figure 1 This is a flowchart of the present invention; Figure 2 This is a model diagram of a planar two-bar linkage spatial robotic arm; Figure 3 Output a curve for the angle of joint 1; Figure 4 Output curves for the angle of joint 2; Figure 5 Output a curve for the angle of joint 1; Figure 6 Output curves for the angle of joint 2. Detailed Implementation

[0017] Specific Implementation Method 1: The specific process of the improved non-singleton interval type-II fuzzy PID spatial robotic arm tracking control method of particle swarm optimization in this implementation method is as follows:

[0018] Step 1: Establish a flexible joint spatial robotic arm model that takes into account faults;

[0019] Determine if the space robotic arm is faulty. If not, proceed to steps two through four; if it is faulty, proceed to steps five through six.

[0020] Step 2: Construct a traditional PID controller and a non-singleton interval type-II fuzzy PID controller;

[0021] Step 3: Use the particle swarm optimization algorithm to optimize the parameters ΔK of the non-singleton interval type-II fuzzy PID controller. P24 ΔK I24 ΔK D24 ΔK P25 ΔK I25 ΔK D25 Optimize to obtain the optimal parameters of the non-singleton interval type II fuzzy PID controller, that is, obtain the optimal non-singleton interval type II fuzzy PID controller.

[0022] Step 4: Control the flexible joint spatial manipulator model established in Step 1 based on the optimal non-singleton interval type II fuzzy PID controller;

[0023] Step 5: Construct a traditional PID controller, a non-singleton interval type II fuzzy PID controller, and a signal-compensated fuzzy PID (the signal-compensated fuzzy PID is an FPID);

[0024] Step 6: Based on the traditional PID controller, non-singleton interval type II fuzzy PID controller, and signal-compensated fuzzy PID controller constructed in Step 5, control the flexible joint spatial manipulator model established in Step 1.

[0025] Specific Implementation Method Two: This implementation method differs from Specific Implementation Method One in that: in step one, a flexible joint spatial robotic arm model considering faults is established; the specific process is as follows:

[0026] Assume the spatial robotic arm is unaffected by external forces such as gravity, can be modeled in an inertial frame, and all joints are flexible and rotatable. As shown in Table 1, where the superscript of the vectors indicates the coordinate system, the meanings of the various quantities of the robotic arm are as follows.

[0027] Table 1. Physical quantities and their meanings of the space robotic arm.

[0028]

[0029]

[0030] The design of controllers for robotic arms that move within a certain plane, as applied to this invention, is also of general applicability. This paper will take a planar two-bar flexible joint spatial robotic arm as the controlled object and model it, such as... Figure 2 ;

[0031] The dynamic model of the planar two-bar flexible joint spatial manipulator considering faults is as follows:

[0032]

[0033] In the formula, q represents the generalized coordinates of the robotic arm. Represents the derivative of the generalized coordinates of the robotic arm. The second derivative of the generalized coordinates of the robotic arm;

[0034] D(q) represents the inertia matrix. Let K(q) represent the centrifugal force and Coriolis force matrices, and K(q) represent the stiffness matrix. Represents the fault vector;

[0035] u represents the generalized force of the robotic arm, γ(t) represents the time distribution of the fault, and t represents time;

[0036] The generalized coordinate q and the generalized force u of the robotic arm are as follows:

[0037] q = [r bx r by q bz q m1 q m2 ] T (2)

[0038] u = [F bx F by τ bz τ m1 τ m2 ] T (3)

[0039] In the formula, r bx This indicates the position of the robot arm base in the X direction within the inertial frame; r by Indicates the position of the robot arm base in the Y direction in the inertial frame; q bz This represents the rotation angle of the robot arm base in the Z direction within the inertial frame;

[0040] q m1 Indicates the rotation angle of joint 1; q m2 Indicates the rotation angle of joint 2; the superscript T indicates the transpose.

[0041] F bx F represents the force acting on the robot arm base in the X direction within an inertial frame of reference; by This represents the force acting on the base of the robotic arm in the Y direction within an inertial frame of reference.

[0042] τ bz This represents the torque acting in the Z direction on the base of the robotic arm in an inertial frame of reference.

[0043] τ m1τ represents the torque exerted by the controller on joint 1. m2 This indicates the torque exerted by the controller on joint 2;

[0044] The controlled object, i.e., the robotic arm, receives a generalized force u as input, provided by the controller, and outputs a generalized coordinate q. This invention focuses on the input torque τ of joint 1. m1 Joint 2 input torque τ m2 And the angles and angular velocities of joints 1 and 2: q m1 q m2 ,

[0045] The other steps and parameters are the same as in Specific Implementation Method 1.

[0046] Specific Implementation Method Three: This implementation method differs from Specific Implementation Method One or Two in that step two involves constructing a traditional PID controller and a non-singleton interval type-II fuzzy PID controller; the specific process is as follows:

[0047] Unlike singleton interval type-II fuzzy PID controllers, this invention employs a non-singleton interval type-II fuzzy PID controller, which is more suitable for handling uncertainties. The input signal of the interval type-II fuzzy logic system is the error signal e and its derivative signal.

[0048] The PID controller has 5 channels, namely F bx F by τ bz τ m1 and τ m2 i = 1, 2, 3, 4, 5;

[0049] The traditional PID controller expression is:

[0050]

[0051] The force F acting on the robot arm base in the X direction in an inertial frame bx It is the output of the first channel traditional PID;

[0052] The force F acting on the robot arm base in the Y direction in an inertial frame by It is the output of the second channel traditional PID;

[0053] The torque τ acting on the robot arm base in the Z direction in an inertial frame bz It is the output of the third channel traditional PID;

[0054] The expression for a non-singleton interval type-II fuzzy PID controller is:

[0055]

[0056] The torque τ of the non-singleton interval type II fuzzy controller acting on joint 1 m1 It is the output of the fourth channel fuzzy PID;

[0057] The torque τ of the non-singleton interval type II fuzzy controller acting on joint 2 m2 It is the output of the fifth channel fuzzy PID;

[0058] In the formula,

[0059] K P1 In response to F bx The proportional coefficient of a traditional PID controller; K I1 In response to F bx The integral coefficient of the traditional PID controller; K D1 In response to F bx The derivative coefficients of a traditional PID controller; where K P1 =10, K I1 =1,K D1 =2;

[0060] K P2 In response to F by The proportional coefficient of a traditional PID controller; K I2 In response to F by The integral coefficient of the traditional PID controller; K D2 In response to F by The derivative coefficients of a traditional PID controller; where K P2 =10, K I2 =1,K D2 =2;

[0061] K P3 For τ bz The proportional coefficient of a traditional PID controller; K I3 For τ bz The integral coefficient of the traditional PID controller; K D3 For τ bz The derivative coefficients of a traditional PID controller; where K P3 =11, K I3 =1,K D3 =2.5;

[0062] K P4 For τ m1 The proportional gain of the non-single-instance interval type-II fuzzy PID controller; K I4 For τ m1 Integral coefficients of the non-single-instance interval type-II fuzzy PID controller; K D4 For τ m1The derivative coefficients of the non-singleton interval type-II fuzzy PID controller; where K P4 =16, K I4 =0.5, K D4 =2;

[0063] K P5 For τ m2 The proportional gain of the non-single-instance interval type-II fuzzy PID controller; K I5 For τ m2 Integral coefficients of the non-single-instance interval type-II fuzzy PID controller; K D5 For τ m2 The derivative coefficients of the non-singleton interval type-II fuzzy PID controller; where K P5 =10, K I5 =0.1, K D5 =1.5;

[0064] e1 indicates that F bx The error signal of the traditional PID controller; e1 represents the derivative of F; e2 represents the derivative of F. by The error signal of the traditional PID controller; e2 represents the derivative of τ; e3 represents the derivative of τ. bz The error signal of the traditional PID controller; e3 represents the derivative of τ; e4 represents the derivative of τ. m1 Error signal of non-single-instance interval type-II fuzzy PID controller; e4 represents the derivative of τ; e5 represents the derivative of τ. m2 Error signal of non-single-instance interval type-II fuzzy PID controller; This represents the derivative of e5;

[0065] ΔK P24 This indicates that the sharp value is output by a non-singleton interval type-II fuzzy PID controller, relative to the proportional coefficient K of the PID controller. P4 Perform real-time tuning; ΔK I24 This indicates that the sharp value is output by a non-singleton interval type II fuzzy PID controller with respect to the integral coefficient K of the PID controller. I4 Perform real-time tuning; ΔK D24 This indicates that the sharp value is output by a non-singleton interval type II fuzzy PID controller with respect to the differential coefficient K of the PID controller. D4 Perform real-time adjustment;

[0066] ΔK P25 This indicates that the sharp value is output by a non-singleton interval type-II fuzzy PID controller, relative to the proportional coefficient K of the PID controller. P5 Perform real-time tuning; ΔK I25This indicates that the sharp value is output by a non-singleton interval type II fuzzy PID controller with respect to the integral coefficient K of the PID controller. I5 Perform real-time tuning; ΔK D25 This indicates that the sharp value is output by a non-singleton interval type II fuzzy PID controller with respect to the differential coefficient K of the PID controller. D5 Perform real-time adjustment;

[0067] Non-singleton interval type-two fuzzy PID controller u i F represents bx F by τ bz τ m1 or τ m2 ;

[0068] This invention employs a fault-tolerant control method with interval-type II fuzzy PID signal compensation to address faults. For the error between the faulty system output and the ideal output, the fuzzy PID can promptly compensate the input signal, thereby correcting the system output.

[0069] Other steps and parameters are the same as in specific implementation method one or two.

[0070] Specific Implementation Method Four: This implementation method differs from Specific Implementation Methods One to Three in that: the ΔK... P24 ΔK I24 ΔK D24 ΔK P25 ΔK I25 ΔK D25 The solution process is as follows:

[0071] 1) Select 5 fuzzy subsets: NB, NS, Z, PS, PB;

[0072] NB represents negative large, NS represents negative small, Z represents zero, PS represents positive small, and PB represents positive large;

[0073] Error signal e of non-singleton interval type II fuzzy PID controller i It is divided into 5 fuzzy subsets: NB, NS, Z, PS, and PB; e i Indicates e4 or e5;

[0074] Error signal e of non-singleton interval type II fuzzy PID controller i Differential It is divided into 5 fuzzy subsets: NB, NS, Z, PS, and PB. express or

[0075] Based on the error signal e of the non-singleton interval type-II fuzzy PID controller i and e i Differential Construct 25 fuzzy rules; e4 and Corresponding to 25 fuzzy rules; e5 and Corresponding to 25 fuzzy rules;

[0076] The fuzzy rule is (i represents 4 or 5):

[0077] When the non-single-instance interval type-II fuzzy PID controller system error signal e i For NB, e i Differential When it is NB, K Pi The fuzzy subset is PS, K Ii The fuzzy subset is NB, K Di The fuzzy subset is PB;

[0078] When the non-single-instance interval type-II fuzzy PID controller system error signal e i For NS, e i Differential When it is NB, K Pi The fuzzy subset is PS, K Ii The fuzzy subsets are NS, K Di The fuzzy subset is NS;

[0079] When the non-single-instance interval type-II fuzzy PID controller system error signal e i For Z, e i Differential When it is NB, K Pi The fuzzy subsets are NS, K Ii The fuzzy subsets are NS, K Di The fuzzy subset is NS;

[0080] When the non-single-instance interval type-II fuzzy PID controller system error signal e i For PS, e i Differential When it is NB, K Pi The fuzzy subset is PS, K Ii The fuzzy subsets are NS, K Di The fuzzy subset is NS;

[0081] When the non-single-instance interval type-II fuzzy PID controller system error signal e i For PB, e i Differential When it is NB, K Pi The fuzzy subset is PB, K Ii The fuzzy subsets are Z and K. Di The fuzzy subset is Z;

[0082] When the non-single-instance interval type-II fuzzy PID controller system error signal e i For NB, e i Differential When K is NS, Pi The fuzzy subset is PS, K Ii The fuzzy subset is NB, K Di The fuzzy subset is PS;

[0083] When the non-single-instance interval type-II fuzzy PID controller system error signal e i For NS, e i Differential When K is NS, Pi The fuzzy subsets are Z and K. Ii The fuzzy subsets are NS, K Di The fuzzy subset is Z;

[0084] When the non-single-instance interval type-II fuzzy PID controller system error signal e i For Z, e i Differential When K is NS, Pi The fuzzy subsets are Z and K. Ii The fuzzy subsets are NS, K Di The fuzzy subset is Z;

[0085] When the non-single-instance interval type-II fuzzy PID controller system error signal e i For PS, e i Differential When K is NS, Pi The fuzzy subsets are Z and K. Ii The fuzzy subsets are Z and K. Di The fuzzy subset is Z;

[0086] When the non-single-instance interval type-II fuzzy PID controller system error signal e i For PB, e i Differential When K is NS, Pi The fuzzy subsets are Z and K. Ii The fuzzy subset is PS, K Di The fuzzy subset is Z;

[0087] When the non-single-instance interval type-II fuzzy PID controller system error signal e i For NB, e i Differential When it is Z, K Pi The fuzzy subsets are NS, K Ii The fuzzy subsets are NS, K Di The fuzzy subset is PS;

[0088] When the non-single-instance interval type-II fuzzy PID controller system error signal e i For NS, e i Differential When it is Z, K Pi The fuzzy subsets are Z and K. Ii The fuzzy subsets are NS, K Di The fuzzy subset is PS;

[0089] When the non-single-instance interval type-II fuzzy PID controller system error signal e i For Z, e i Differential When it is Z, K Pi The fuzzy subsets are Z and K. Ii The fuzzy subsets are Z and K. Di The fuzzy subset is Z;

[0090] When the non-single-instance interval type-II fuzzy PID controller system error signal e i For PS, e i Differential When it is Z, K Pi The fuzzy subsets are Z and K. Ii The fuzzy subset is PS, K Di The fuzzy subset is PS;

[0091] When the non-single-instance interval type-II fuzzy PID controller system error signal e i For PB, e i Differential When it is Z, K Pi The fuzzy subsets are NS, K Ii The fuzzy subset is PS, K Di The fuzzy subset is PS;

[0092] When the non-single-instance interval type-II fuzzy PID controller system error signal e i For NB, e i Differential When it is PS, K Pi The fuzzy subsets are Z and K. Ii The fuzzy subsets are NS, K Di The fuzzy subset is Z;

[0093] When the non-single-instance interval type-II fuzzy PID controller system error signal e i For NS, e i Differential When it is PS, K Pi The fuzzy subsets are Z and K. Ii The fuzzy subsets are Z and K. Di The fuzzy subset is Z;

[0094] When the non-single-instance interval type-II fuzzy PID controller system error signal e i For Z, e i Differential When it is PS, K Pi The fuzzy subsets are Z and K. Ii The fuzzy subset is PS, K Di The fuzzy subset is Z;

[0095] When the non-single-instance interval type-II fuzzy PID controller system error signal e i For PS, e i Differential When it is PS, K Pi The fuzzy subsets are Z and K. Ii The fuzzy subset is PS, K Di The fuzzy subset is Z;

[0096] When the non-single-instance interval type-II fuzzy PID controller system error signal e i For PB, e i Differential When it is PS, K Pi The fuzzy subset is PS, K Ii The fuzzy subset is PB, K Di The fuzzy subset is PS;

[0097] When the non-single-instance interval type-II fuzzy PID controller system error signal e i For NB, e i Differential When K is PB, Pi The fuzzy subset is PB, K Ii The fuzzy subsets are Z and K. Di The fuzzy subset is Z;

[0098] When the non-single-instance interval type-II fuzzy PID controller system error signal e i For NS, e i Differential When K is PB, Pi The fuzzy subset is PS, K Ii The fuzzy subset is PS, K Di The fuzzy subset is NS;

[0099] When the non-single-instance interval type-II fuzzy PID controller system error signal e i For Z, e i Differential When K is PB, Pi The fuzzy subsets are NS, K Ii The fuzzy subset is PS, K Di The fuzzy subset is NS;

[0100] When the non-single-instance interval type-II fuzzy PID controller system error signal e i For PS, e i Differential When K is PB, Pi The fuzzy subset is PS, K Ii The fuzzy subset is PS, K Di The fuzzy subset is NS;

[0101] When the non-single-instance interval type-II fuzzy PID controller system error signal e i For PB, e i Differential When K is PB, Pi The fuzzy subset is PS, K Ii The fuzzy subset is PB, K Di The fuzzy subset is PB;

[0102] The fuzzy rules for interval type II fuzzy PID are shown in the table below. The fuzzy inference input is the system error and its derivative, denoted by e in the table. i , This indicates that, taking the bottom right cell of the table as an example, the fuzzy logic for that cell is as follows:

[0103] Table 2. Fuzzy Rules for KP / KI / KD

[0104]

[0105] Because of the characteristics of the controlled object having multiple inputs and multiple outputs, this invention uses a 5-channel PID controller, and the basic parameters are shown in Table 3.

[0106] Table 3 Basic parameters of the fuzzy PID controller

[0107]

[0108] 2) Let For the input of a non-singleton interval type-II fuzzy PID controller, the trigger interval F of the j-th fuzzy rule is... j (x) is

[0109]

[0110] Among them, e i This is the error signal for a non-single-instance interval type-II fuzzy PID controller. For e i The differential; j = 1, 2, ..., 25; e i Indicates e4 or e5; express or

[0111] The upper bound of the trigger interval The Lower World f j for:

[0112]

[0113] In the formula, For τ m1 The j-th fuzzy rule corresponds to the fuzzy subset (K corresponding to the 25 rules). Pi K Ii K Di A fuzzy subset, such as PS / PB / PB), For τ m2 The j-th fuzzy rule corresponds to a fuzzy subset. Taking the bottom right cell of Table 3 as an example, for the j-th fuzzy rule,

[0114] Indicates that e i The j-th fuzzy rule corresponds to the lower bound membership function value of the fuzzy subset. Indicating targeting The j-th fuzzy rule corresponds to the lower bound membership function value of the fuzzy subset;

[0115] Indicates that e i The j-th fuzzy rule corresponds to the upper bound membership function value of the fuzzy subset. Indicating targeting The j-th fuzzy rule corresponds to the upper bound membership function value of the fuzzy subset;

[0116] 3) Output interval y of the j-th fuzzy rule j for:

[0117]

[0118] in, y j This is the lower bound of the output interval for the fuzzy subset. The upper bound of the output interval for the fuzzy subset;

[0119] Upper bound of the output interval of fuzzy subset NB upper bound of the output interval of fuzzy subset NS Upper bound of the output interval of the fuzzy subset Z Upper bound of the output interval of fuzzy subset PS Upper bound of the output interval of fuzzy subset PB

[0120] Lower bound of output interval for fuzzy subset NB y j=-1; Lower bound of the output interval of the fuzzy subset NS y j = -0.6; Lower bound of the output interval for the fuzzy subset Z y j = -0.1; Lower bound of the output interval of the fuzzy subset PS y j =0.4; Lower bound of the output interval of fuzzy subset PB y j =0.9;

[0121] The output ranges of the five fuzzy subsets are shown in the table below, where y j To output the lower bound of the fuzzy subset output interval, Output the upper bound of the interval for a type II fuzzy subset of the interval.

[0122] Table 4 Output Range

[0123]

[0124] 4) Based on the upper bound of the trigger interval The Lower World f j as well as y j , Calculate the lower bound set center y l and the upper bound set center y u ; indicates as:

[0125]

[0126] Where l and u are the transition points obtained through the KM iteration method; n = 25;

[0127] 5) Based on the lower bound set center y l and the upper bound set center y u Calculate the lower bound set center y l and the upper bound set center y u The average value, i.e., the clarity value of the final output;

[0128]

[0129] 6) The clarity value and K of the final output obtained based on 5). P4 , obtain ΔK P24 ; indicates as:

[0130]

[0131] in, For τ m1 About K P4 Scale factor;

[0132] For τ m1 About K P4 The final output is the clear value y;

[0133] Based on the clarity value and K of the final output obtained in step 5) I4 , obtain ΔK I24 ; indicates as:

[0134]

[0135] in, For τ m1 About K I4 Integral factors;

[0136] For τ m1 About K I4 The final output is the clear value y;

[0137] Based on the clarity value and K of the final output obtained in step 5) D4 , obtain ΔK D24 ; indicates as:

[0138]

[0139] in,

[0140] For τ m1 About K D4 Integral factors;

[0141] For τ m1 About K D4 The final output is the clear value y;

[0142] 7) The clarity value and K of the final output obtained based on 5). P5 , obtain ΔK P25 ; indicates as:

[0143]

[0144] in, For τ m2 About K P5 Scale factor;

[0145] For τ m2 About K P5 The final output is the clear value y;

[0146] Based on the clarity value and K of the final output obtained in step 5)I5 , obtain ΔK I25 ; indicates as:

[0147]

[0148] in, For τ m2 About K I5 Scale factor;

[0149] For τ m2 About K P5 The final output is the clear value y;

[0150] Based on the clarity value and K of the final output obtained in step 5) D5 , obtain ΔK D25 ; indicates as:

[0151]

[0152] in, For τ m2 About K D5 Integral factors;

[0153] For τ m2 About K P5 The final output is the clear value y;

[0154] Different values;

[0155] This invention selects five fuzzy subsets: NB, NS, Z, PS, and PB, and adopts the membership function of a base-width isosceles triangle, as shown in the following formula:

[0156]

[0157]

[0158] Where x is the independent variable of the function, A i Let i be a certain fuzzy subset, i = 1, 2, 3, 4, 5, corresponding to NB, NS, Z, PS, PB respectively; For A i The upper bound membership function output value; For A i Lower bound membership function output value; x i Let x be the x-coordinate of the center of the isosceles triangle; a and b are constants, where a and b represent the width of the isosceles triangle, satisfying b > a > 0.

[0159] The other steps and parameters are the same as those in one of the specific implementation methods one to three.

[0160] Specific Implementation Method Five: This implementation method differs from Specific Implementation Methods One through Four in that: in step three, a particle swarm optimization algorithm is used to optimize the parameter ΔK of the non-singleton interval type-II fuzzy PID controller. P24 ΔK I24 ΔK D24 ΔK P25 ΔK I25 ΔK D25 Optimization is performed (optimizing all 6 parameters together) to obtain the optimal parameters for the non-singleton interval type II fuzzy PID controller, thus obtaining the optimal non-singleton interval type II fuzzy PID controller; the specific process is as follows:

[0161] This invention employs a particle swarm optimization algorithm to optimize the parameters of an interval type-II fuzzy PID controller, thereby obtaining more suitable parameters ΔK for a non-singleton interval type-II fuzzy PID controller. P24 ΔK I24 ΔK D24 ΔK P25 ΔK I25 ΔK D25 The specific parameters to be optimized are the 4-channel and 5-channel blur scaling factors, totaling 6 parameters. Therefore, in this particle swarm optimization algorithm, the particle position is a 6-dimensional vector.

[0162] First, multiple particles are generated within a certain range. Each particle has two basic attributes: position and velocity. Its initial position and initial velocity are randomly generated. Then, the search for the optimal solution begins iteratively until the search target is reached.

[0163] In the particle swarm optimization algorithm, the inertia coefficient w decreases with the number of iterations, as shown in the following equation:

[0164]

[0165] In the formula, i is the iteration number, i max w represents the maximum number of iterations. min For the minimum inertia coefficient, w max The maximum inertia coefficient;

[0166] The inertia coefficient w in the traditional particle swarm optimization algorithm is a constant. This invention improves this by enhancing the algorithm's global search capability in the early stage and its local search capability in the later stage.

[0167] The fitness calculation method in the particle swarm optimization algorithm is as follows:

[0168]

[0169] In the formula, |e1(t)| is the absolute value of the angle error of joint 1; |e2(t)| is the absolute value of the angle error of joint 2; t is time; T1 is the upper bound time of integration; T2 is the upper bound time of integration; T2 < T1; α1, α2, β1, β2 are weighting factors; J is the particle fitness;

[0170] The particle fitness attribute is calculated from the particle position. For traditional particle swarm optimization systems, the performance evaluation index ITAE is generally set directly equal to the particle fitness J; the smaller the value of this index, the better the system performance. This fitness calculation method helps reduce system settling time and overshoot, but because the error has a smaller impact on ITAE when time t is small, the system's speed decreases. Therefore, this invention proposes a new fitness calculation method to simultaneously consider system accuracy and speed.

[0171] The other steps and parameters are the same as those in specific implementation methods one through four.

[0172] Specific Implementation Method Six: This implementation method differs from Specific Implementation Methods One through Five in that step five involves constructing a traditional PID controller, a non-singleton interval type-II fuzzy PID controller, and a signal-compensated fuzzy PID controller (the signal-compensated fuzzy PID controller is an FPID controller); the specific process is as follows:

[0173] The fuzzy PID expression for signal compensation is:

[0174]

[0175] Signal compensation, fuzzy PID compensation, q m1 It is the output of the first channel fuzzy PID;

[0176]

[0177] Signal compensation, fuzzy PID compensation, q m2 It is the output of the second channel fuzzy PID;

[0178] In the formula, q m1 Indicates the rotation angle of joint 1; q m2 Indicates the rotation angle of joint 2;

[0179] K PF1 For q m1 The signal compensation fuzzy PID proportional coefficient; K IF1 For q m1 Signal compensation fuzzy PID integral coefficients; K DF1 For q m1 The signal compensation fuzzy PID differential coefficients; where K PF1 =5,K IF1 =0.1, K DF1=0.01;

[0180] K PF2 For q m2 The signal compensation fuzzy PID proportional coefficient; K IF2 For q m2 Signal compensation fuzzy PID integral coefficients; K DF2 For q m2 The signal compensation fuzzy PID differential coefficients; where K PF2 =5,K IF2 =0.1, K DF2 =0.01;

[0181] e F1 Indicates that for q m1 The signal compensation for fuzzy PID error signal; e F1 The differential;

[0182] e F2 Indicates that for q m2 The signal compensation for fuzzy PID error signal; e F2 The differential;

[0183] ΔK PF1 This indicates that the fuzzy PID output is a clear value due to signal compensation, and the proportional coefficient K of the PID controller is affected. PF1 Perform real-time tuning; ΔK IF1 This indicates that the fuzzy PID output is a clear value due to signal compensation, and the integral coefficient K of the PID controller is used to express this value. IF1 Perform real-time tuning; ΔK DF1 This indicates that the fuzzy PID output is a clear value due to signal compensation, with respect to the differential coefficient K of the PID controller. DF1 Perform real-time adjustment;

[0184] ΔK PF2 This indicates that the fuzzy PID output is a clear value due to signal compensation, and the proportional coefficient K of the PID controller is affected. PF2 Perform real-time tuning; ΔK IF2 This indicates that the fuzzy PID output is a clear value due to signal compensation, and the integral coefficient K of the PID controller is used to express this value. IF2 Perform real-time tuning; ΔK DF2 This indicates that the fuzzy PID output is a clear value due to signal compensation, with respect to the differential coefficient K of the PID controller. DF2 Perform real-time adjustment;

[0185] The parameters are shown in Table 5. The present invention has designed fuzzy rules specifically for them, as shown in Table 6.

[0186] Table 5. Signal Compensation Fuzzy PID Parameters

[0187]

[0188] The other steps and parameters are the same as those in specific implementation methods one through five.

[0189] Specific Implementation Method Seven: This implementation method differs from Specific Implementation Methods One through Six in that: the ΔK... PF1 ΔK IF1 ΔK DF1 ΔK PF2 ΔK IF2 ΔK DF2 The solution process is as follows:

[0190] 1) Select 5 fuzzy subsets: NB, NS, Z, PS, PB;

[0191] NB represents negative large, NS represents negative small, Z represents zero, PS represents positive small, and PB represents positive large;

[0192] Regarding q m1 Signal compensation for fuzzy PID error signal e F1 Divided into 5 fuzzy subsets: NB, NS, Z, PS, and PB; targeting e F1 Differential It is divided into 5 fuzzy subsets: NB, NS, Z, PS, and PB.

[0193] Regarding q m2 Signal compensation for fuzzy PID error signal e F2 Divided into 5 fuzzy subsets: NB, NS, Z, PS, and PB; targeting e F2 Differential It is divided into 5 fuzzy subsets: NB, NS, Z, PS, and PB.

[0194] Based on Signal compensation for fuzzy PID error signal and Differential Construct 25 fuzzy rules;

[0195] q m1 or q m2 ; e F1 or e F2 ; express or

[0196] Based on q m1 Signal compensation for fuzzy PID error signal e F1 and e F1 Differential Construct 25 fuzzy rules;

[0197] Based on q m2 Signal compensation for fuzzy PID error signal e F2 and e F2 Differential Construct 25 fuzzy rules;

[0198] The fuzzy rule is ( (Indicates 1 or 2):

[0199] When targeting Signal compensation for fuzzy PID error signal For NB, Differential When it is NB, The fuzzy subset is PB. The fuzzy subset is NB. The fuzzy subset is PB;

[0200] K represents PF1 or K PF2 ; K represents IF1 or K IF2 ; K represents DF1 or K DF2 When targeting Signal compensation for fuzzy PID error signal For NB, targeting Differential When it is NS, The fuzzy subset is PS. The fuzzy subset is NS. The fuzzy subset is NS;

[0201] When targeting Signal compensation for fuzzy PID error signal For NB, targeting Differential When it is Z, The fuzzy subset is PS. The fuzzy subset is NS. The fuzzy subset is NS;

[0202] When targeting Signal compensation for fuzzy PID error signal For NB, targeting Differential When using PS, The fuzzy subset is PS. The fuzzy subset is NS. The fuzzy subset is NS;

[0203] When targeting Signal compensation for fuzzy PID error signal For NB, targeting Differential When it is PB, The fuzzy subset is Z. The fuzzy subset is Z. The fuzzy subset is Z;

[0204] When targeting Signal compensation for fuzzy PID error signal For NS, targeting Differential When it is NB, The fuzzy subset is PS. The fuzzy subset is NB. The fuzzy subset is PS;

[0205] When targeting Signal compensation for fuzzy PID error signal For NS, targeting Differential When it is NS, The fuzzy subset is PS. The fuzzy subset is NS. The fuzzy subset is Z;

[0206] When targeting Signal compensation for fuzzy PID error signal For NS, targeting Differential When it is Z, The fuzzy subset is PS. The fuzzy subset is NS. The fuzzy subset is Z;

[0207] When targeting Signal compensation for fuzzy PID error signal For NS, targeting Differential When using PS, The fuzzy subset is Z. The fuzzy subset is Z. The fuzzy subset is Z;

[0208] When targeting Signal compensation for fuzzy PID error signal For NS, targeting Differential When it is PB, The fuzzy subset is NS. The fuzzy subset is PS. The fuzzy subset is Z;

[0209] When targeting Signal compensation for fuzzy PID error signal For Z, targeting Differential When it is NB, The fuzzy subset is PS, K IF1 The fuzzy subset is NS. The fuzzy subset is PS;

[0210] When targeting Signal compensation for fuzzy PID error signal For Z, targeting Differential When it is NS, The fuzzy subset is PS. The fuzzy subset is NS. The fuzzy subset is PS;

[0211] When targeting Signal compensation for fuzzy PID error signal For Z, targeting Differential When it is Z, The fuzzy subset is Z. The fuzzy subset is Z. The fuzzy subset is Z;

[0212] When targeting Signal compensation for fuzzy PID error signal For Z, targeting Differential When using PS, The fuzzy subset is NS. The fuzzy subset is PS. The fuzzy subset is PS;

[0213] When targeting Signal compensation for fuzzy PID error signal For Z, targeting Differential When it is PB, The fuzzy subset is NS. The fuzzy subset is PS. The fuzzy subset is PS;

[0214] When targeting Signal compensation for fuzzy PID error signal For Photoshop, targeting Differential When it is NB, The fuzzy subset is PS. The fuzzy subset is NS. The fuzzy subset is Z;

[0215] When targeting Signal compensation for fuzzy PID error signal For Photoshop, targeting Differential When it is NS, The fuzzy subset is Z. The fuzzy subset is Z. The fuzzy subset is Z;

[0216] When targeting Signal compensation for fuzzy PID error signal For Photoshop, targeting Differential When it is Z, The fuzzy subset is NS. The fuzzy subset is PS. The fuzzy subset is Z;

[0217] When targeting Signal compensation for fuzzy PID error signal For Photoshop, targeting Differential When using PS, The fuzzy subset is NS. The fuzzy subset is PS. The fuzzy subset is Z;

[0218] When targeting Signal compensation for fuzzy PID error signal For Photoshop, targeting Differential When it is PB, The fuzzy subset is NS. The fuzzy subset is PB. The fuzzy subset is PS;

[0219] When targeting Signal compensation for fuzzy PID error signal For PB, targeting Differential When it is NB, The fuzzy subset is Z. The fuzzy subset is Z. The fuzzy subset is Z;

[0220] When targeting Signal compensation for fuzzy PID error signal For PB, targeting Differential When it is NS, The fuzzy subset is NS. The fuzzy subset is PS. The fuzzy subset is NS;

[0221] When targeting Signal compensation for fuzzy PID error signal For PB, targeting Differential When it is Z, The fuzzy subset is NS. The fuzzy subset is PS. The fuzzy subset is NS;

[0222] When targeting Signal compensation for fuzzy PID error signal For PB, targeting Differential When using PS, The fuzzy subset is NS. The fuzzy subset is PS. The fuzzy subset is NS;

[0223] When targeting Signal compensation for fuzzy PID error signal For PB, the differential is... When it is PB, The fuzzy subset is NB. The fuzzy subset is PB. The fuzzy subset is NB;

[0224] Table 6. Rules for Fuzzy PID Signal Compensation (KPF / KIF / KDF)

[0225]

[0226] 2) Let For signal compensation fuzzy PID input, the trigger interval F of the j-th fuzzy rule j (x) is

[0227]

[0228] in, To compensate for fuzzy PID error signals; To compensate for the fuzzy PID error signal e i The differential; j = 1, 2, ..., 25; eF1 or e F2 ; express or

[0229] The upper bound of the trigger interval The Lower World f j for:

[0230]

[0231] In the formula, For q m1 The j-th fuzzy rule corresponds to the fuzzy subset (K corresponding to the 25 rules). Pi K Ii K Di A fuzzy subset, such as PS / PB / PB), For q m2 The j-th fuzzy rule corresponds to a fuzzy subset. Taking the bottom right cell of Table 3 as an example, for the j-th fuzzy rule,

[0232] Indicating targeting The j-th fuzzy rule corresponds to the lower bound membership function value of the fuzzy subset.

[0233] Indicating targeting The j-th fuzzy rule corresponds to the lower bound membership function value of the fuzzy subset;

[0234] Indicating targeting The j-th fuzzy rule corresponds to the upper bound membership function value of the fuzzy subset.

[0235] Indicating targeting The j-th fuzzy rule corresponds to the upper bound membership function value of the fuzzy subset;

[0236] 3) Output interval y of the j-th fuzzy rule j for:

[0237]

[0238] in, y j This is the lower bound of the output interval for the fuzzy subset. The upper bound of the output interval for the fuzzy subset;

[0239] Upper bound of the output interval of fuzzy subset NB upper bound of the output interval of fuzzy subset NS Upper bound of the output interval of the fuzzy subset Z Upper bound of the output interval of fuzzy subset PS Upper bound of the output interval of fuzzy subset PB

[0240] Lower bound of output interval for fuzzy subset NB y j =-1; Lower bound of the output interval of the fuzzy subset NS y j = -0.6; Lower bound of the output interval for the fuzzy subset Z y j = -0.1; Lower bound of the output interval of the fuzzy subset PS y j =0.4; Lower bound of the output interval of fuzzy subset PB y j =0.9;

[0241] The output ranges of the five fuzzy subsets are shown in the table below, where y j To output the lower bound of the fuzzy subset output interval, This is the upper bound of the output interval for the fuzzy subset.

[0242] Table 7 Output Range

[0243]

[0244] 4) Based on the upper bound of the trigger interval The Lower World f j as well as y j , Calculate the lower bound set center y l and the upper bound set center y u ; indicates as:

[0245]

[0246] Where l and u are the transition points obtained through the KM iteration method; n = 25;

[0247] 5) Based on the lower bound set center y l and the upper bound set center y u Calculate the lower bound set center y l and the upper bound set center y u The average value, i.e., the clarity value of the final output;

[0248]

[0249] 6) The clarity value and K of the final output obtained based on 5). PF1 , obtain ΔK PF1 ; indicates as:

[0250]

[0251] in, For q m1 About K PF1 Scale factor;

[0252] For q m1 About K PF1 The final output is the clear value y;

[0253] Based on the clarity value and K of the final output obtained in step 5) IF1 , obtain ΔK IF1 ; indicates as:

[0254]

[0255] in, For q m1 About K IF1 Integral factors;

[0256] For q m1 About K IF1 The final output is the clear value y;

[0257] Based on the clarity value and K of the final output obtained in step 5) DF1 , obtain ΔK DF1 ; indicates as:

[0258]

[0259] in, For q m1 About K DF1 Integral factors;

[0260] For q m1 About K DF1 The final output is the clear value y;

[0261] 7) The clarity value and K of the final output obtained based on 5). P5 , obtain ΔK PF2 ; indicates as:

[0262]

[0263] in, For q m2 About K P5 Scale factor;

[0264] For q m2 About K P5 The final output is the clear value y;

[0265] Based on the clarity value and K of the final output obtained in step 5) IF2 , obtain ΔK IF2 ; indicates as:

[0266]

[0267] in, For q m2 About K IF2 Scale factor;

[0268] For q m2 About K IF2 The final output is the clear value y;

[0269] Based on the clarity value and K of the final output obtained in step 5) DF2 , obtain ΔK DF2 ; indicates as:

[0270]

[0271] in, For q m2 About K DF2 Integral factors;

[0272] For q m2 About K DF2 The final output is the clear value y;

[0273] This invention selects five fuzzy subsets: NB, NS, Z, PS, and PB, and adopts the membership function of a base-width isosceles triangle, as shown in the following formula:

[0274]

[0275]

[0276] Where x is the independent variable of the function, A i Let i be a certain fuzzy subset, i = 1, 2, 3, 4, 5, corresponding to NB, NS, Z, PS, PB respectively; For A i The upper bound membership function output value; For A i Lower bound membership function output value; x i Let x be the x-coordinate of the center of the isosceles triangle; a and b are constants, where a and b represent the width of the isosceles triangle, satisfying b > a > 0.

[0277] The other steps and parameters are the same as those in one of the specific implementation methods one to six.

[0278] The beneficial effects of the present invention are verified using the following embodiments:

[0279] Example 1:

[0280] To verify and demonstrate the effectiveness of the improved non-singleton interval type-two fuzzy PID controller and the interval type-two fuzzy PID signal compensation fault-tolerant control in this invention, the following simulation experiments were conducted. The robotic arm parameters are shown in Table 8, and the interval type-two fuzzy membership function parameters are a = 0.25, b = 0.5, x... i See Table 9.

[0281] Table 8. Specific values ​​of various physical quantities of the two-bar linkage robotic arm.

[0282]

[0283] Table 9. Parameters of Interval Type II Fuzzy Membership Function

[0284]

[0285] First, assume no faults occur, and input the desired generalized coordinates q = [0 0 -1 1 1]. T Run the simulation program and compare the output curves, such as Figure 3 , Figure 4 “PSO” indicates particle swarm optimization, and “S” and “N” represent traditional singleton fuzzification and non-singleton fuzzification, respectively. The control performance metrics are shown in the table below.

[0286] Table 10 Performance Indicators of Fault-Free Systems

[0287]

[0288] As can be seen from the output curves, the unoptimized interval type-two fuzzy PID controller exhibits relatively poor performance with a large overshoot, making it difficult to meet the stability requirements. The improved particle swarm optimization algorithm for the interval type-two fuzzy PID controller mitigates this shortcoming, enabling the system to meet the task requirements. Table 10 shows that the non-singleton fuzzy system outperforms the singleton fuzzy system in terms of performance indicators and control effect.

[0289] Add an actuator fault τ to both joints of the robotic arm at t=4s. mi (t)=τ mi (t)+1, i=1,2. Input expected generalized coordinates q=[0 0 -1 1 1] T Run the simulation program and compare the output curves, such as Figure 5 , Figure 6“SC” indicates that a fuzzy PID signal compensation fault-tolerant control method is used. The control performance indicators are shown in the table below.

[0290] Table 11 Performance Indicators of Faulty Systems

[0291]

[0292] When a malfunction occurs in the joint actuator of a robotic arm, the joint angle of a system without fuzzy PID signal compensation is difficult to return to the expected value, while the system with fuzzy PID signal compensation is less affected by the malfunction. Furthermore, for the faulty system, the performance indicators of the non-singleton fuzzy system are better than those of the singleton fuzzy system, indicating that the non-singleton fuzzy PID has a stronger ability to handle malfunctions than the traditional singleton fuzzy PID. This demonstrates that the controller designed in this invention has good robustness and can better handle malfunctions during the performance of tasks by a flexible joint spatial robotic arm.

[0293] This invention may have other embodiments. Without departing from the spirit and essence of this invention, those skilled in the art can make various corresponding changes and modifications according to this invention, but these corresponding changes and modifications should all fall within the protection scope of the appended claims.

Claims

1. An improved particle swarm optimization non-singleton interval type-II fuzzy PID spatial robotic arm tracking control method, characterized by: The specific process of the method is as follows: Step 1: Establish a model of a flexible joint spatial robotic arm; Determine if the space robotic arm is faulty. If not, proceed to steps two through four; if it is faulty, proceed to steps five through six. Step 2: Construct a traditional PID controller and a non-singleton interval type-II fuzzy PID controller; Step 3: Use the particle swarm optimization algorithm to optimize the parameters of the non-singleton interval type II fuzzy PID controller to obtain the optimal non-singleton interval type II fuzzy PID controller parameters, that is, obtain the optimal non-singleton interval type II fuzzy PID controller. Step 4: Control the flexible joint spatial manipulator model established in Step 1 based on the optimal non-singleton interval type II fuzzy PID controller; Step 5: Construct a traditional PID controller, a non-singleton interval type-II fuzzy PID controller, and a signal-compensated fuzzy PID controller; Step 6: Based on the traditional PID controller, non-singleton interval type II fuzzy PID controller and signal-compensated fuzzy PID controller constructed in Step 5, control the flexible joint spatial manipulator model established in Step 1. Step two involves constructing a traditional PID controller and a non-singleton interval type-II fuzzy PID controller; the specific process is as follows: The traditional PID controller expression is: The force F acting on the robot arm base in the X direction in an inertial frame bx It is the output of the first channel traditional PID; The force F acting on the robot arm base in the Y direction in an inertial frame by It is the output of the second channel traditional PID; The torque τ acting on the robot arm base in the Z direction in an inertial frame bz It is the output of the third channel traditional PID; The expression for a non-singleton interval type-II fuzzy PID controller is: The torque τ of the non-singleton interval type II fuzzy controller acting on joint 1 m1 It is the output of the fourth channel fuzzy PID; The torque τ of the non-singleton interval type II fuzzy controller acting on joint 2 m2 It is the output of the fifth channel fuzzy PID; In the formula, K P1 In response to F bx The proportional coefficient of a traditional PID controller; K I1 In response to F bx The integral coefficient of the traditional PID controller; K D1 In response to F bx The derivative coefficients of a traditional PID controller; where K P1 =10, K I1 =1,K D1 =2; K P2 In response to F by The proportional coefficient of a traditional PID controller; K I2 In response to F by The integral coefficient of the traditional PID controller; K D2 In response to F by The derivative coefficients of a traditional PID controller; where K P2 =10, K I2 =1,K D2 =2; K P3 For τ bz The proportional coefficient of a traditional PID controller; K I3 For τ bz The integral coefficient of the traditional PID controller; K D3 For τ bz The derivative coefficients of a traditional PID controller; where K P3 =11, K I3 =1,K D3 =2.5; K P4 For τ m1 The proportional gain of the non-single-instance interval type-II fuzzy PID controller; K I4 For τ m1 Integral coefficients of the non-single-instance interval type-II fuzzy PID controller; K D4 For τ m1 The derivative coefficients of the non-singleton interval type-II fuzzy PID controller; where K P4 =16, K I4 =0.5, K D4 =2; K P5 For τ m2 The proportional gain of the non-single-instance interval type-II fuzzy PID controller; K I5 For τ m2 Integral coefficients of the non-single-instance interval type-II fuzzy PID controller; K D5 For τ m2 The derivative coefficients of the non-singleton interval type-II fuzzy PID controller; where K P5 =10, K I5 =0.1, K D5 =1.5; e1 indicates that F bx The error signal of the traditional PID controller; e1 represents the derivative of F; e2 represents the derivative of F. by The error signal of the traditional PID controller; e2 represents the derivative of τ; e3 represents the derivative of τ. bz The error signal of the traditional PID controller; e3 represents the derivative of τ; e4 represents the derivative of τ. m1 Error signal of non-single-instance interval type-II fuzzy PID controller; e4 represents the derivative of τ; e5 represents the derivative of τ. m2 Error signal of non-single-instance interval type-II fuzzy PID controller; This represents the derivative of e5; ΔK P24 This indicates that the sharp value is output by a non-singleton interval type II fuzzy PID controller; ΔK I24 This indicates that the sharp value is output by a non-singleton interval type II fuzzy PID controller; ΔK D24 This indicates that the sharp value is output by a non-singleton interval type II fuzzy PID controller; ΔK P25 This indicates that the sharp value is output by a non-singleton interval type II fuzzy PID controller; ΔK I25 This indicates that the sharp value is output by a non-singleton interval type II fuzzy PID controller; ΔK D25 This indicates that the output is a clear value from a non-singleton interval type II fuzzy PID controller.

2. The improved particle swarm optimization non-singleton interval type-II fuzzy PID spatial robotic arm tracking control method according to claim 1, characterized in that: The first step involves establishing a flexible joint spatial robotic arm model; the specific process is as follows: The dynamic model of the planar two-bar flexible joint spatial manipulator considering faults is as follows: In the formula, q represents the generalized coordinate of the robotic arm. Represents the derivative of the generalized coordinates of the robotic arm. The second derivative of the generalized coordinates of the robotic arm; D(q) represents the inertia matrix. Let K(q) represent the centrifugal force and Coriolis force matrices, and K(q) represent the stiffness matrix. Represents the fault vector; u represents the generalized force of the robotic arm, γ(t) represents the time distribution of the fault, and t represents time; The generalized coordinate q and the generalized force u of the robotic arm are as follows: q=[r bx r by q bz q m1 q m2 ] T (2) u=[F bx F by t bz t m1 t m2 ] T (3) In the formula, r bx This indicates the position of the robot arm base in the X direction within the inertial frame; r by Indicates the position of the robot arm base in the Y direction in the inertial frame; q bz This represents the rotation angle of the robot arm base in the Z direction within the inertial frame; q m1 Indicates the rotation angle of joint 1; q m2 Indicates the rotation angle of joint 2; The superscript T indicates transpose; F bx F represents the force acting on the robot arm base in the X direction within an inertial frame of reference; by This represents the force acting on the base of the robotic arm in the Y direction within an inertial frame of reference. τ bz This represents the torque acting in the Z direction on the base of the robotic arm in an inertial frame of reference. τ m1 τ represents the torque exerted by the controller on joint 1. m2 This indicates the torque exerted by the controller on joint 2.

3. The improved particle swarm optimization non-singleton interval type-II fuzzy PID spatial robotic arm tracking control method according to claim 2, characterized in that: The ΔK P24 ΔK I24 ΔK D24 ΔK P25 ΔK I25 ΔK D25 The solution process is as follows: 1) Select 5 fuzzy subsets: NB, NS, Z, PS, PB; NB represents negative large, NS represents negative small, Z represents zero, PS represents positive small, and PB represents positive large; Error signal e of non-singleton interval type II fuzzy PID controller i It is divided into 5 fuzzy subsets: NB, NS, Z, PS, and PB; e i Indicates e4 or e5; Error signal e of non-singleton interval type II fuzzy PID controller i Differential It is divided into 5 fuzzy subsets: NB, NS, Z, PS, and PB. express or Based on the error signal e of the non-singleton interval type-II fuzzy PID controller i and e i Differential Construct 25 fuzzy rules; The fuzzy rule is: When the non-single-instance interval type-II fuzzy PID controller system error signal e i For NB, e i Differential When it is NB, K Pi The fuzzy subset is PS, K Ii The fuzzy subset is NB, K Di The fuzzy subset is PB; When the non-single-instance interval type-II fuzzy PID controller system error signal e i For NS, e i Differential When it is NB, K Pi The fuzzy subset is PS, K Ii The fuzzy subsets are NS, K Di The fuzzy subset is NS; When the non-single-instance interval type-II fuzzy PID controller system error signal e i For Z, e i Differential When it is NB, K Pi The fuzzy subsets are NS, K Ii The fuzzy subsets are NS, K Di The fuzzy subset is NS; When the non-single-instance interval type-II fuzzy PID controller system error signal e i For PS, e i Differential When it is NB, K Pi The fuzzy subset is PS, K Ii The fuzzy subsets are NS, K Di The fuzzy subset is NS; When the non-single-instance interval type-II fuzzy PID controller system error signal e i For PB, e i Differential When it is NB, K Pi The fuzzy subset is PB, K Ii The fuzzy subsets are Z and K. Di The fuzzy subset is Z; When the non-single-instance interval type-II fuzzy PID controller system error signal e i For NB, e i Differential When K is NS, Pi The fuzzy subset is PS, K Ii The fuzzy subset is NB, K Di The fuzzy subset is PS; When the non-single-instance interval type-II fuzzy PID controller system error signal e i For NS, e i Differential When K is NS, Pi The fuzzy subsets are Z and K. Ii The fuzzy subsets are NS, K Di The fuzzy subset is Z; When the non-single-instance interval type-II fuzzy PID controller system error signal e i For Z, e i Differential When K is NS, Pi The fuzzy subsets are Z and K. Ii The fuzzy subsets are NS, K Di The fuzzy subset is Z; When the non-single-instance interval type-II fuzzy PID controller system error signal e i For PS, e i Differential When K is NS, Pi The fuzzy subsets are Z and K. Ii The fuzzy subsets are Z and K. Di The fuzzy subset is Z; When the non-single-instance interval type-II fuzzy PID controller system error signal e i For PB, e i Differential When K is NS, Pi The fuzzy subsets are Z and K. Ii The fuzzy subset is PS, K Di The fuzzy subset is Z; When the non-single-instance interval type-II fuzzy PID controller system error signal e i For NB, e i Differential When it is Z, K Pi The fuzzy subsets are NS, K Ii The fuzzy subsets are NS, K Di The fuzzy subset is PS; When the non-single-instance interval type-II fuzzy PID controller system error signal e i For NS, e i Differential When it is Z, K Pi The fuzzy subsets are Z and K. Ii The fuzzy subsets are NS, K Di The fuzzy subset is PS; When the non-single-instance interval type-II fuzzy PID controller system error signal e i For Z, e i Differential When it is Z, K Pi The fuzzy subsets are Z and K. Ii The fuzzy subsets are Z and K. Di The fuzzy subset is Z; When the non-single-instance interval type-II fuzzy PID controller system error signal e i For PS, e i Differential When it is Z, K Pi The fuzzy subsets are Z and K. Ii The fuzzy subset is PS, K Di The fuzzy subset is PS; When the non-single-instance interval type-II fuzzy PID controller system error signal e i For PB, e i Differential When it is Z, K Pi The fuzzy subsets are NS, K Ii The fuzzy subset is PS, K Di The fuzzy subset is PS; When the non-single-instance interval type-II fuzzy PID controller system error signal e i For NB, e i Differential When it is PS, K Pi The fuzzy subsets are Z and K. Ii The fuzzy subsets are NS, K Di The fuzzy subset is Z; When the non-single-instance interval type-II fuzzy PID controller system error signal e i For NS, e i Differential When it is PS, K Pi The fuzzy subsets are Z and K. Ii The fuzzy subsets are Z and K. Di The fuzzy subset is Z; When the non-single-instance interval type-II fuzzy PID controller system error signal e i For Z, e i Differential When it is PS, K Pi The fuzzy subsets are Z and K. Ii The fuzzy subset is PS, K Di The fuzzy subset is Z; When the non-single-instance interval type-II fuzzy PID controller system error signal e i For PS, e i Differential When it is PS, K Pi The fuzzy subsets are Z and K. Ii The fuzzy subset is PS, K Di The fuzzy subset is Z; When the non-single-instance interval type-II fuzzy PID controller system error signal e i For PB, e i Differential When it is PS, K Pi The fuzzy subset is PS, K Ii The fuzzy subset is PB, K Di The fuzzy subset is PS; When the non-single-instance interval type-II fuzzy PID controller system error signal e i For NB, e i Differential When K is PB, Pi The fuzzy subset is PB, K Ii The fuzzy subsets are Z and K. Di The fuzzy subset is Z; When the non-single-instance interval type-II fuzzy PID controller system error signal e i For NS, e i Differential When K is PB, Pi The fuzzy subset is PS, K Ii The fuzzy subset is PS, K Di The fuzzy subset is NS; When the non-single-instance interval type-II fuzzy PID controller system error signal e i For Z, e i Differential When K is PB, Pi The fuzzy subsets are NS, K Ii The fuzzy subset is PS, K Di The fuzzy subset is NS; When the non-single-instance interval type-II fuzzy PID controller system error signal e i For PS, e i Differential When K is PB, Pi The fuzzy subset is PS, K Ii The fuzzy subset is PS, K Di The fuzzy subset is NS; When the non-single-instance interval type-II fuzzy PID controller system error signal e i For PB, e i Differential When K is PB, Pi The fuzzy subset is PS, K Ii The fuzzy subset is PB, K Di The fuzzy subset is PB; 2) Let For the input of a non-singleton interval type-II fuzzy PID controller, the trigger interval F of the j-th fuzzy rule is... j (x) is Among them, e i This is the error signal for a non-single-instance interval type-II fuzzy PID controller. For e i The differential; j = 1, 2, ..., 25; e i Indicates e4 or e5; express or The upper bound of the trigger interval The Lower World f j for: In the formula, For τ m1 The j-th fuzzy rule corresponds to the fuzzy subset. For τ m2 The j-th fuzzy rule corresponds to the fuzzy subset; Indicates that e i The j-th fuzzy rule corresponds to the lower bound membership function value of the fuzzy subset. Indicating targeting The j-th fuzzy rule corresponds to the lower bound membership function value of the fuzzy subset; Indicates that e i The j-th fuzzy rule corresponds to the upper bound membership function value of the fuzzy subset. Indicating targeting The j-th fuzzy rule corresponds to the upper bound membership function value of the fuzzy subset; 3) Output interval y of the j-th fuzzy rule j for: in, y j This is the lower bound of the output interval for the fuzzy subset. The upper bound of the output interval for the fuzzy subset; Upper bound of the output interval of fuzzy subset NB upper bound of the output interval of fuzzy subset NS Upper bound of the output interval of the fuzzy subset Z Upper bound of the output interval of fuzzy subset PS Upper bound of the output interval of fuzzy subset PB Lower bound of output interval for fuzzy subset NB y j =-1; Lower bound of the output interval of the fuzzy subset NS y j = -0.6; Lower bound of the output interval for the fuzzy subset Z y j = -0.1; Lower bound of the output interval of the fuzzy subset PS y j =0.4; Lower bound of the output interval of the fuzzy subset PB y j =0.9; 4) Based on the upper bound of the trigger interval The Lower World f j as well as y j , Calculate the lower bound set center y l and the upper bound set center y u ; indicates as: Where l and u are the transition points obtained through the KM iteration method; n = 25; 5) Based on the lower bound set center y l and the upper bound set center y u Calculate the lower bound set center y l and the upper bound set center y u The average value, i.e., the clarity value of the final output; 6) The clarity value and K of the final output obtained based on 5). P4 , obtain ΔK P24 ; indicates as: in, For τ m1 About K P4 Scale factor; For τ m1 About K P4 The final output is the clear value y; Based on the clarity value and K of the final output obtained in step 5) I4 , obtain ΔK I24 ; indicates as: in, For τ m1 About K I4 Integral factors; For τ m1 About K I4 The final output is the clear value y; Based on the clarity value and K of the final output obtained in step 5) D4 , obtain ΔK D24 ; indicates as: in, For τ m1 About K D4 Integral factors; For τ m1 About K D4 The final output is the clear value y; 7) The clarity value and K of the final output obtained based on 5). P5 , obtain ΔK P25 ; indicates as: in, For τ m2 About K P5 Scale factor; For τ m2 About K P5 The final output is the clear value y; Based on the clarity value and K of the final output obtained in step 5) I5 , obtain ΔK I25 ; indicates as: in, For τ m2 About K I5 Scale factor; For τ m2 About K P5 The final output is the clear value y; Based on the clarity value and K of the final output obtained in step 5) D5 , obtain ΔK D25 ; indicates as: in, For τ m2 About K D5 Integral factors; For τ m2 About K P5 The final output is the clear value y.

4. The improved particle swarm optimization non-singleton interval type-II fuzzy PID spatial robotic arm tracking control method according to claim 3, characterized in that: In step three, the particle swarm optimization algorithm is used to optimize the parameters of the non-singleton interval type II fuzzy PID controller to obtain the optimal non-singleton interval type II fuzzy PID controller parameters, that is, to obtain the optimal non-singleton interval type II fuzzy PID controller. The specific process is as follows: In the particle swarm optimization algorithm, the inertia coefficient w decreases with the number of iterations, as shown in the following equation: In the formula, i is the iteration number, i max w represents the maximum number of iterations. min For the minimum inertia coefficient, w max The maximum inertia coefficient; The fitness calculation method in the particle swarm optimization algorithm is as follows: In the formula, |e1(t)| is the absolute value of the angle error of joint 1; |e2(t)| is the absolute value of the angle error of joint 2; t is time; T1 is the upper bound time of integration; T2 is the upper bound time of integration; T2 < T1; α1, α2, β1, β2 are weighting factors; J is the particle fitness.

5. The improved particle swarm optimization non-singleton interval type-II fuzzy PID spatial robotic arm tracking control method according to claim 4, characterized in that: Step five involves constructing a traditional PID controller, a non-singleton interval type-II fuzzy PID controller, and a signal-compensated fuzzy PID controller; the specific process is as follows: The fuzzy PID expression for signal compensation is: Signal compensation, fuzzy PID compensation, q m1 It is the output of the first channel fuzzy PID; Signal compensation, fuzzy PID compensation, q m2 It is the output of the second channel fuzzy PID; In the formula, q m1 Indicates the rotation angle of joint 1; q m2 Indicates the rotation angle of joint 2; K PF1 For q m1 The signal compensation fuzzy PID proportional coefficient; K IF1 For q m1 Signal compensation fuzzy PID integral coefficients; K DF1 For q m1 The signal compensation fuzzy PID differential coefficients; where K PF1 =5,K IF1 =0.1, K DF1 =0.01; K PF2 For q m2 The signal compensation fuzzy PID proportional coefficient; K IF2 For q m2 Signal compensation fuzzy PID integral coefficients; K DF2 For q m2 The signal compensation fuzzy PID differential coefficients; where K PF2 =5,K IF2 =0.1, K DF2 =0.01; e F1 Indicates that for q m1 The signal compensation for fuzzy PID error signal; e F1 The differential; e F2 Indicates that for q m2 The signal compensation for fuzzy PID error signal; e F2 The differential; ΔK PF1 This indicates that the fuzzy PID output is sharpened by signal compensation; ΔK IF1 This indicates that the fuzzy PID output is sharpened by signal compensation; ΔK DF1 This indicates that the fuzzy PID output is cleared by signal compensation; ΔK PF2 This indicates that the fuzzy PID output is sharpened by signal compensation; ΔK IF2 This indicates that the fuzzy PID output is sharpened by signal compensation; ΔK DF2 This indicates that the fuzzy PID output is cleared by signal compensation.

6. The improved particle swarm optimization non-singleton interval type-II fuzzy PID spatial robotic arm tracking control method according to claim 5, characterized in that: The ΔK PF1 ΔK IF1 ΔK DF1 ΔK PF2 ΔK IF2 ΔK DF2 The solution process is as follows: 1) Select 5 fuzzy subsets: NB, NS, Z, PS, PB; NB represents negative large, NS represents negative small, Z represents zero, PS represents positive small, and PB represents positive large; Regarding q m1 Signal compensation for fuzzy PID error signal e F1 Divided into 5 fuzzy subsets: NB, NS, Z, PS, and PB; targeting e F1 Differential It is divided into 5 fuzzy subsets: NB, NS, Z, PS, and PB. Regarding q m2 Signal compensation for fuzzy PID error signal e F2 Divided into 5 fuzzy subsets: NB, NS, Z, PS, and PB; targeting e F2 Differential It is divided into 5 fuzzy subsets: NB, NS, Z, PS, and PB. Based on Signal compensation for fuzzy PID error signal and Differential Construct 25 fuzzy rules; q m1 or q m2 ; e F1 or e F2 ; express or The fuzzy rule is: When targeting Signal compensation for fuzzy PID error signal For NB, Differential When it is NB, The fuzzy subset is PB. The fuzzy subset is NB. The fuzzy subset is PB; K represents PF1 or K PF2 ; K represents IF1 or K IF2 ; K represents DF1 or K DF2 ; When targeting Signal compensation for fuzzy PID error signal For NB, targeting Differential When it is NS, The fuzzy subset of is PS. The fuzzy subset is NS. The fuzzy subset is NS; When targeting Signal compensation for fuzzy PID error signal For NB, targeting Differential When it is Z, The fuzzy subset of is PS. The fuzzy subset is NS. The fuzzy subset is NS; When targeting Signal compensation for fuzzy PID error signal For NB, targeting Differential When using PS, The fuzzy subset of is PS. The fuzzy subset is NS. The fuzzy subset is NS; When targeting Signal compensation for fuzzy PID error signal For NB, targeting Differential When it is PB, The fuzzy subset is Z. The fuzzy subset is Z. The fuzzy subset is Z; When targeting Signal compensation for fuzzy PID error signal For NS, targeting Differential When it is NB, The fuzzy subset of is PS. The fuzzy subset is NB. The fuzzy subset is PS; When targeting Signal compensation for fuzzy PID error signal For NS, targeting Differential When it is NS, The fuzzy subset of is PS. The fuzzy subset is NS. The fuzzy subset is Z; When targeting Signal compensation for fuzzy PID error signal For NS, targeting Differential When it is Z, The fuzzy subset of is PS. The fuzzy subset is NS. The fuzzy subset is Z; When targeting Signal compensation for fuzzy PID error signal For NS, targeting Differential When using PS, The fuzzy subset is Z. The fuzzy subset is Z. The fuzzy subset is Z; When targeting Signal compensation for fuzzy PID error signal For NS, targeting Differential When it is PB, The fuzzy subset is NS. The fuzzy subset of is PS. The fuzzy subset is Z; When targeting Signal compensation for fuzzy PID error signal For Z, targeting Differential When it is NB, The fuzzy subset is PS, K IF1 The fuzzy subset is NS. The fuzzy subset is PS; When targeting Signal compensation for fuzzy PID error signal For Z, targeting Differential When it is NS, The fuzzy subset of is PS. The fuzzy subset is NS. The fuzzy subset is PS; When targeting Signal compensation for fuzzy PID error signal For Z, targeting Differential When it is Z, The fuzzy subset is Z. The fuzzy subset is Z. The fuzzy subset is Z; When targeting Signal compensation for fuzzy PID error signal For Z, targeting Differential When using PS, The fuzzy subset is NS. The fuzzy subset of is PS. The fuzzy subset is PS; When targeting Signal compensation for fuzzy PID error signal For Z, targeting Differential When it is PB, The fuzzy subset is NS. The fuzzy subset of is PS. The fuzzy subset is PS; When targeting Signal compensation for fuzzy PID error signal For Photoshop, targeting Differential When it is NB, The fuzzy subset of is PS. The fuzzy subset is NS. The fuzzy subset is Z; When targeting Signal compensation for fuzzy PID error signal For Photoshop, targeting Differential When it is NS, The fuzzy subset is Z. The fuzzy subset is Z. The fuzzy subset is Z; When targeting Signal compensation for fuzzy PID error signal For Photoshop, targeting Differential When it is Z, The fuzzy subset is NS. The fuzzy subset of is PS. The fuzzy subset is Z; When targeting Signal compensation for fuzzy PID error signal For Photoshop, targeting Differential When using PS, The fuzzy subset is NS. The fuzzy subset of is PS. The fuzzy subset is Z; When targeting Signal compensation for fuzzy PID error signal For Photoshop, targeting Differential When it is PB, The fuzzy subset is NS. The fuzzy subset is PB. The fuzzy subset is PS; When targeting Signal compensation for fuzzy PID error signal For PB, targeting Differential When it is NB, The fuzzy subset is Z. The fuzzy subset is Z. The fuzzy subset is Z; When targeting Signal compensation for fuzzy PID error signal For PB, targeting Differential When it is NS, The fuzzy subset is NS. The fuzzy subset of is PS. The fuzzy subset is NS; When targeting Signal compensation for fuzzy PID error signal For PB, targeting Differential When it is Z, The fuzzy subset is NS. The fuzzy subset of is PS. The fuzzy subset is NS; When targeting Signal compensation for fuzzy PID error signal For PB, targeting Differential When using PS, The fuzzy subset is NS. The fuzzy subset of is PS. The fuzzy subset is NS; When targeting Signal compensation for fuzzy PID error signal For PB, the differential is... When it is PB, The fuzzy subset is NB. The fuzzy subset is PB. The fuzzy subset is NB; 2) Let For signal compensation fuzzy PID input, the trigger interval F of the j-th fuzzy rule j (x) is in, To compensate for fuzzy PID error signals; To compensate for the fuzzy PID error signal e i The differential; j = 1, 2, ..., 25; e F1 or e F2 ; express or The upper bound of the trigger interval The Lower World f j for: In the formula, For q m1 The j-th fuzzy rule corresponds to the fuzzy subset. For q m2 The j-th fuzzy rule corresponds to the fuzzy subset; Indicating targeting The j-th fuzzy rule corresponds to the lower bound membership function value of the fuzzy subset. Indicating targeting The j-th fuzzy rule corresponds to the lower bound membership function value of the fuzzy subset; Indicating targeting The j-th fuzzy rule corresponds to the upper bound membership function value of the fuzzy subset. Indicating targeting The j-th fuzzy rule corresponds to the upper bound membership function value of the fuzzy subset; 3) Output interval y of the j-th fuzzy rule j for: in, y j This is the lower bound of the output interval for the fuzzy subset. The upper bound of the output interval for the fuzzy subset; upper bound of the output interval of fuzzy subset NB Upper bound of fuzzy subset NS output interval Upper bound of the output interval of the fuzzy subset Z Upper bound of the output interval of fuzzy subset PS Upper bound of fuzzy subset PB output interval Lower bound of output interval for fuzzy subset NB y j =-1; Lower bound of the output interval of the fuzzy subset NS y j = -0.6; Lower bound of the output interval for the fuzzy subset Z y j = -0.1; Lower bound of the output interval of the fuzzy subset PS y j =0.4; Lower bound of the output interval of the fuzzy subset PB y j =0.9; 4) Based on the upper bound of the trigger interval The Lower World f j as well as y j , Calculate the lower bound set center y l and the upper bound set center y u ; indicates as: Where l and u are the transition points obtained through the KM iteration method; n = 25; 5) Based on the lower bound set center y l and the upper bound set center y u Calculate the lower bound set center y l and the upper bound set center y u The average value, i.e., the clarity value of the final output; 6) The clarity value and K of the final output obtained based on 5). PF1 , obtain ΔK PF1 ; indicates as: in, For q m1 About K PF1 Scale factor; For q m1 About K PF1 The final output is the clear value y; Based on the clarity value and K of the final output obtained in step 5) IF1 , obtain ΔK IF1 ; indicates as: in, For q m1 About K IF1 Integral factors; For q m1 About K IF1 The final output is the clear value y; Based on the clarity value and K of the final output obtained in step 5) DF1 , obtain ΔK DF1 ; indicates as: in, For q m1 About K DF1 Integral factors; For q m1 About K DF1 The final output is the clear value y; 7) The clarity value and K of the final output obtained based on 5). P5 , obtain ΔK PF2 ; indicates as: in, For q m2 About K P5 Scale factor; For q m2 About K P5 The final output is the clear value y; Based on the clarity value and K of the final output obtained in step 5) IF2 , obtain ΔK IF2 ; indicates as: in, For q m2 About K IF2 Scale factor; For q m2 About K IF2 The final output is the clear value y; Based on the clarity value and K of the final output obtained in step 5) DF2 , obtain ΔK DF2 ; indicates as: in, For q m2 About K DF2 Integral factors; For q m2 About K DF2 The final output is the clear value y.

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