Adaptive robust control method for redundant robots with fused differential homeomorphism mapping

Abstract

The present application relates to heavy load robot control technical field, specifically, it is a kind of adaptive robust control method of heavy load robot fusing differential homeomorphism.The present application is by real-time acquisition each joint position information, and accurate comparison is carried out with expected trajectory, exports three input vectors, real-time updates differential homeomorphism parameters by T-S fuzzy reasoning system, and based on the parameters after optimization, joint space state is mapped to new coordinate space, generates collaborative control item through distributed communication to obtain neighbor joint information, constructs sliding surface to generate robust control item in combination with local tracking error, and generates compensation item based on adaptive law parameter uncertainty estimation, forms three parallel control outputs, carries out dynamic weight distribution and fusion to three control items, obtains basic control torque, then superimposes feedforward compensation item based on nominal dynamic model, generates final comprehensive control torque, and the comprehensive control torque guarantees performance while optimizing energy distribution.

Description

Technical Field

[0001] This invention relates to the field of heavy-duty robot control technology, and more specifically, to an adaptive robust control method for heavy-duty robots that incorporates differential homeomorphism mapping. Background Technology

[0002] Differential homeomorphism is a powerful tool in modern control theory and nonlinear system analysis. For example, imagine a piece of clay with a coordinate system (with many grid lines) drawn on it. We can stretch, squeeze, or bend this clay arbitrarily, but we cannot tear it (disrupt continuity) or glue its different points together (ensure reversibility). This deformation process itself is a mapping that uniquely maps every point in the original shape (original coordinate system) to a point in the new shape (new coordinate system). Differential homeomorphism is a smooth and reversible transformation that establishes a one-to-one correspondence between points in two spaces (or two coordinate systems), and this relationship is smooth.

[0003] Heavy-duty welding robots face challenges such as strong coupling between multiple joints, sudden load changes (e.g., from 300kg to 600kg), external disturbances (e.g., impact loads), and parameter perturbations (e.g., joint wear) during operation. These challenges make it difficult to maintain control accuracy under complex conditions, leading to insufficient robustness of the control system. While feedback linearization methods based on differential homeomorphism mapping can effectively handle nonlinear problems, they generally employ fixed parameter mapping and cannot adaptively adjust according to the working state, resulting in decreased control performance. Furthermore, existing control architectures often employ centralized control and lack effective distributed coordination mechanisms, making it difficult to optimize the dynamic coupling between multiple joints, thus limiting system reliability and scalability. Therefore, we propose an adaptive robust control method for heavy-duty robots that integrates differential homeomorphism mapping. Summary of the Invention

[0004] The purpose of this invention is to provide an adaptive robust control method for heavy-duty robots that incorporates differential homeomorphism mapping, in order to solve any of the technical problems raised in the background art.

[0005] To achieve the above objectives, this invention provides an adaptive robust control method for a heavy-duty robot that integrates differential homeomorphism mapping, comprising the following steps:

[0006] S10. Collect the actual position, speed and welding torch load torque information of each joint of the heavy-duty welding robot, compare it with the expected trajectory and output a three-dimensional input vector composed of position tracking error, speed tracking error and load torque.

[0007] S20. Using the ternary input vector as input, update the differential homeomorphism mapping parameters through TS fuzzy inference. Based on these parameters, map the joint space state to the new coordinate space, and simultaneously output the state, velocity, and Jacobian matrix of the new coordinate space. The mapping process is as follows:

[0008] The joint position is transformed into a new coordinate space position through the decoupling matrix of the kernel function and the associated mapping parameters. The joint velocity, combined with the rate of change of the mapping parameters and the coupling relationship between position and velocity, is transformed into a new coordinate space velocity through the combined action of the kernel function derivative and the kernel function.

[0009] S30. In the new coordinate space, establish a distributed collaborative, robust control, and adaptive compensation linkage generation mechanism:

[0010] The cooperative control term is calculated based on the state information of neighboring joints, and its output is used as a feedforward signal. A sliding surface is constructed in combination with the local tracking error. The boundary layer is adaptively adjusted according to the mapping parameters output by fuzzy inference to generate a robust control term. At the same time, based on the changing trend of the sliding surface and the cooperative consistency error, a compensation term is estimated online through an adaptive law to generate a total of three parallel and cooperative control outputs.

[0011] S40. A Nash game equilibrium strategy is adopted to perform multi-objective dynamic weight allocation and fusion of the three control terms. The basic control torque is obtained through Jacobi matrix inverse mapping. Then, combined with feedforward dynamic compensation, the final comprehensive control torque is generated and output to each joint actuator.

[0012] As a further improvement to this technical solution, the steps in S10 for calculating the position tracking error and speed tracking error of each joint of the heavy-duty welding robot are as follows:

[0013] S11. Real-time acquisition of joint rotation angle signals, decoding by the controller to obtain the actual position of each joint, differential operation on the continuously sampled actual position signals to obtain the actual speed of each joint, and connecting a piezoelectric torque sensor in series at the drive end of each joint to acquire the joint output torque in real time.

[0014] S12. Retrieve the preset joint expected trajectory data from the motion controller of the welding robot, extract the expected position and expected speed of the joint at each moment, compare the actual position and actual speed of each joint with the expected position and expected speed of the joint at each moment, and obtain the position tracking error and speed tracking error of each joint of the heavy-duty welding robot.

[0015] S13. Normalize the position tracking error, speed tracking error, and load torque to form a standardized mapping parameter adjustment input vector.

[0016] As a further improvement to this technical solution, S12, based on the task requirements of the heavy-duty welding robot, presets the desired operation trajectory in Cartesian space in the robot motion controller, uses the robot forward kinematics inverse algorithm to convert the desired operation trajectory in Cartesian space into the angular space trajectory of each joint, obtains the desired joint position at each time, performs the first-order differential operation in the time domain on the desired joint position to obtain the desired joint velocity, and retrieves the preset desired joint trajectory data.

[0017] As a further improvement to this technical solution, the specific steps of S20 in updating the differential homeomorphism mapping parameters through TS fuzzy inference are as follows:

[0018] S21. Perform fuzzy set partitioning on the normalized ternary input vector, design membership function based on welding scenario characteristics, and construct fuzzy rule base based on welding process and experimental data.

[0019] S22. For the normalized input variables acquired in real time, calculate their membership degree to each fuzzy set according to the membership function, quantify the degree to which the input variables belong to each fuzzy state, screen out activation rules with a membership degree greater than 0, and perform a weighted average of the outputs of all activation rules to obtain the normalized correction amount.

[0020] S23. Based on the mapping parameters of the previous moment, the amplitude-limited correction amount is superimposed to obtain the mapping parameters of the current moment, ensuring that the updated parameters are within the preset range, so as to maintain the reversibility and decoupling effect of the mapping.

[0021] As a further improvement to this technical solution, when S22 performs a weighted average of the outputs of all activation rules, the activation intensity of the activation rule is calculated by the product method, that is, the product of the membership degrees of the input variables, which reflects the comprehensive matching degree between the input state and the rule antecedents, and the activation intensity of the rule is used as the weight.

[0022] As a further improvement to this technical solution, the specific steps for S30 to calculate the cooperative control term, robust control term, and adaptive compensation term are as follows:

[0023] S31. Define the joint neighbor relationship through distributed communication topology, calculate the state consistency error between this joint and neighbor joints, and combine it with the expected trajectory tracking requirements to form the total cooperative deviation. Synthesize the cooperative control term through the proportional-derivative control structure.

[0024] S32. Calculate the tracking error in the new coordinate space, including position error and velocity error, and construct a linear sliding surface. Combine the mapping parameters output by TS fuzzy inference to design a boundary layer thickness with adaptive characteristics and generate a robust control term.

[0025] S33. The parameter uncertainty is linearized and processed. Based on the changing trend of the sliding surface and the consistency error of the cooperative control term, the parameter estimate is updated online through the adaptive law to generate the adaptive compensation term.

[0026] As a further improvement to this technical solution, S32 uses a saturation function instead of a sign function to divide the range of sliding surface error into two regions: inside and outside the boundary layer. Response characteristics are designed for each region. Outside the boundary layer, the saturation function maintains a step output consistent with the sign function. Inside the boundary layer, the saturation function no longer provides a step output but instead provides a smooth linear output.

[0027] As a further improvement to this technical solution, S32 constructs a comprehensive coordination state evaluation index based on the output amplitude index and the rate of change index of the coordinated control term. Based on the numerical range of the comprehensive coordination state evaluation index, the coordination state is divided into four levels, and a corresponding fourth-order gain adjustment strategy is established. Based on this index, the fourth-order gain adjustment strategy is implemented to dynamically adjust the gain of the robust control term.

[0028] As a further improvement to this technical solution, the specific steps for S40 to generate the final integrated control torque are as follows:

[0029] S41. Treat the cooperative control term, robust sliding mode control term, and adaptive compensation term as three game participants that cooperate and constrain each other. Design a unique cost function for each participant. Through the constraint optimization conditions, derive the dynamic adjustment rules of each weight, and finally obtain the optimal weight combination that satisfies the equilibrium conditions.

[0030] S42. Based on the dynamic weights obtained from the Nash equilibrium solution, the three control terms are weighted and summed to obtain the comprehensive control input of the new coordinate space. Using the bridge established by the Jacobian matrix, the comprehensive control input of the new coordinate space is inversely mapped to the original joint space to obtain the basic control torque of the joint space.

[0031] S43. Based on the basic control torque, a feedforward compensation term based on the nominal dynamics model is superimposed to form a comprehensive control torque.

[0032] As a further improvement to this technical solution, when the Jacobian matrix is ​​singular, S42 uses a least-squares pseudo-inverse algorithm with a damping factor to perform inverse transformation solution. The damping factor is dynamically adjusted based on the singular value spectrum of the Jacobian matrix. By calculating the condition number and the minimum singular value of the matrix in real time, a nonlinear mapping relationship between the damping factor and the ill-conditioned degree of the matrix is ​​established, minimizing the joint moment norm and limiting the spatial gain of the solution.

[0033] Compared with the prior art, the beneficial effects of the present invention are as follows:

[0034] 1. This adaptive robust control method for a heavy-duty robot, which integrates differential homeomorphism mapping, collects real-time information on the actual position, velocity, and welding torch load torque of each joint and compares it precisely with the desired trajectory. It outputs a three-dimensional input vector containing position tracking error, velocity tracking error, and load torque, establishing a precise foundation for working condition identification and providing a reliable input basis for subsequent intelligent control. Through real-time error calculation, it ensures the system's rapid response capability to dynamic changes.

[0035] 2. Based on the ternary input vector, the differential homeomorphism mapping parameters are updated in real time through the TS fuzzy inference system. Based on the optimized parameters, the joint space state is mapped to the new coordinate space, and the state, velocity, and Jacobian matrix of the new coordinate space are output synchronously. Through adaptive adjustment of the mapping parameters, the system can automatically optimize the transformation characteristics according to different operating conditions. The differential homeomorphism mapping effectively decouples the nonlinear characteristics of the system and greatly simplifies the subsequent control design. The real-time calculation of the Jacobian matrix provides a precise mathematical basis for the inverse transformation of the control force.

[0036] 3. In the new coordinate space, the system acquires neighbor joint information through distributed communication to generate cooperative control terms, constructs a sliding surface based on local tracking errors to generate robust control terms, and generates compensation terms based on adaptive law to estimate parameter uncertainties, forming three parallel control outputs. Distributed cooperative control ensures the synchronization and coordination of multi-joint motion; robust sliding mode control provides strong anti-disturbance capability; adaptive compensation realizes online learning and cancellation of parameter uncertainties; the parallel generation and linkage coordination of the three controls form a complete control and protection system.

[0037] 4. A Nash game equilibrium strategy is adopted to dynamically allocate and fuse the three control terms. The basic control torque is obtained through the inverse transformation of the Jacobian matrix, and then a feedforward compensation term based on the nominal dynamics model is superimposed to generate the final comprehensive control torque. The Nash game equilibrium realizes intelligent trade-offs and optimization of multiple control objectives. The damping factor design of the pseudo-inverse algorithm ensures the stability of the control in the singular region. The dynamic feedforward compensation improves the system's response speed and tracking accuracy. The final comprehensive control torque optimizes energy distribution while ensuring performance.

[0038] In addition to the objectives, features, and advantages described above, the present invention has other objectives, features, and advantages. The invention will now be described in further detail with reference to the figures. Attached Figure Description

[0039] Figure 1 This is a schematic diagram of the overall method flow of the present invention. Detailed Implementation

[0040] The technical solutions in 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.

[0041] Currently, heavy-duty welding robots face challenges such as strong coupling between multiple joints, sudden load changes (e.g., from 300kg to 600kg), external disturbances (e.g., impact loads), and parameter perturbations (e.g., joint wear) during operation. These challenges make it difficult to maintain control accuracy under complex working conditions, resulting in insufficient robustness of the control system. While feedback linearization methods based on differential homeomorphism mapping can effectively handle nonlinear problems, they generally use fixed parameter mapping and cannot adaptively adjust according to the working state, leading to a decline in control performance. In addition, existing control architectures mostly adopt centralized control and lack effective distributed coordination mechanisms, making it difficult to optimize the dynamic coupling between multiple joints, thus limiting system reliability and scalability.

[0042] Therefore, this invention proposes an adaptive robust control method for a heavy-load robot that integrates differential homeomorphism mapping, specifically including:

[0043] The ternary input vector generated in step S10 provides comprehensive operating condition characteristics for the fuzzy inference system in step S20. The position error reflects the steady-state accuracy requirement, the speed error reflects the dynamic response characteristics, and the load torque characterizes the changes in external operating conditions. The three together guide the optimization direction of the mapping parameters.

[0044] The adaptive differential homeomorphism mapping obtained through step S20 provides an ideal working platform for the multimodal control generation in step S30. The new coordinate space after mapping has the characteristics of approximate linearity and weak coupling, which greatly improves the efficiency and performance of subsequent control design. In step S30, although the cooperative control term, robust control term and adaptive compensation term are generated in parallel, they form a tight coupling relationship through information sharing.

[0045] The three parallel control outputs generated in step S30 provide rich decision options for the Nash game fusion in S40. Each control output represents the control requirements of the system in different dimensions. The cooperative control term reflects the multi-joint coordination requirements, the robust control term reflects the disturbance rejection requirements, and the adaptive compensation term represents the parameter learning requirements.

[0046] Please refer to the details below. Figure 1 As shown, it includes the following steps:

[0047] First, by collecting the actual position, speed and load torque information of each joint in real time and comparing it with the expected trajectory, a standardized three-dimensional input vector is formed, thereby solving the problem of incomplete and inaccurate state perception in traditional control and providing a reliable basis for the system to perceive the working conditions.

[0048] Secondly, by adopting TS fuzzy inference and adaptive differential homeomorphism mapping technology, the three-element input vector is used as input, the mapping parameters are updated in real time, and the joint space state is transformed to the new coordinate space. The state, velocity and Jacobian matrix of the new coordinate space are output synchronously. This effectively solves the control design problem caused by the strong nonlinearity and strong coupling characteristics of heavy-duty robots. Through the intelligent adjustment of adaptive mapping parameters, the linearization simplification of complex systems is realized, creating favorable conditions for subsequent control design.

[0049] Then, three control outputs are generated in parallel in the new coordinate space. The information of neighboring joints is obtained and the cooperative control terms are calculated through distributed cooperative control technology. Robust control terms are generated by combining sliding surface construction and adaptive boundary layer technology. At the same time, parameter uncertainty is estimated online based on adaptive law and compensation terms are generated. Through the parallel generation and linkage of multimodal control, the complex control problems such as multi-joint motion step loss, external disturbance suppression and parameter uncertainty compensation are effectively solved, forming a control guarantee system with complementary functions.

[0050] Finally, a Nash game equilibrium strategy is adopted to dynamically allocate and fuse the three control terms. The control input in the new coordinate space is inversely mapped to the joint space through the damped least squares pseudo-inverse algorithm. Combined with model-based feedforward compensation to generate the final control torque, the conflict trade-off problem among multiple control objectives is solved, ensuring the stable control of the system in the singular region and achieving optimization and improvement of the overall performance.

[0051] The steps in S10 for calculating the position tracking error and velocity tracking error of each joint of the heavy-duty welding robot are as follows:

[0052] S11. Real-time acquisition of joint rotation angle signals, decoding by the controller to obtain the actual position of each joint, differential operation on the continuously sampled actual position signals to obtain the actual speed of each joint, and connecting a piezoelectric torque sensor in series at the drive end of each joint to acquire the joint output torque in real time.

[0053] The encoder collects the rotation angle signal in real time, and the controller decodes and outputs the actual position of each joint. The encoder outputs digital rotation angle signal (Gray code or binary code) at a fixed frequency, which is transmitted to the distributed controller via bus. The controller has a built-in encoder decoding module that converts the digital signal into an absolute angle value.

[0054] By performing time-domain difference analysis on continuously sampled position signals and combining it with filtering, a stable actual joint velocity is obtained. Based on the continuously sampled position signals, the joint velocity is calculated using the time-domain difference method. The core idea is to reflect the motion rate through the position difference between adjacent time points. The formula is as follows:

[0055] ;

[0056] in, For joint velocity, for The position of the joints at all times. This represents the position of the joint at the previous moment. To control the cycle.

[0057] Piezoelectric torque sensors are connected in series at the drive ends of each joint to collect the output torque in real time and transmit it synchronously to the controller. Piezoelectric static torque sensors are selected and installed in series between the output end of the joint reducer and the connecting rod of the robotic arm. The two ends of the sensor are rigidly connected by flanges to ensure that the torque is transmitted without leakage. The piezoelectric sensor outputs a weak charge signal, which is converted into a voltage signal by the built-in charge amplifier and then converted into a digital signal by the A / D converter.

[0058] The above steps are used to collect information on the actual position, speed, and welding torch load torque of each joint of the heavy-duty welding robot, providing real-time data for comparison with the desired trajectory.

[0059] S12. Retrieve the preset joint expected trajectory data from the motion controller of the welding robot, extract the expected position and expected speed of the joint at each moment, compare the actual position and actual speed of each joint with the expected position and expected speed of the joint at each moment, and obtain the position tracking error and speed tracking error of each joint of the heavy-duty welding robot.

[0060] The preset desired trajectory is generated by the weld path planning module and stored in the trajectory buffer of the motion controller. It contains joint motion commands for the entire welding operation. The controller synchronously retrieves the desired data corresponding to the current time k according to the control cycle to ensure complete alignment with the actual state sampling time and the desired position of each joint. Expected speed ;

[0061] Based on the comparison between the actual value and the expected value at the same moment and the same joint, the error is directly calculated, and the position tracking error is:

[0062] ;

[0063] in, for Real-time joint position tracking error, for The position of the joints at all times. for The desired position of the joint at any given moment.

[0064] The speed tracking error is:

[0065] ;

[0066] in, for The speed tracking error of the joint at any given moment. for The speed of the joint at all times, for The expected speed of the joint at any given moment.

[0067] In order to better obtain the expected joint position and expected velocity at each moment, S12, based on the task requirements of the heavy-duty welding robot, presets the expected operation trajectory in Cartesian space in the robot motion controller, uses the robot forward kinematics inverse algorithm to convert the expected operation trajectory in Cartesian space into the angular space trajectory of each joint, obtains the expected joint position at each moment, performs the first-order differential operation in the time domain on the expected joint position to obtain the expected joint velocity, and retrieves the preset expected joint trajectory data.

[0068] Based on the welding task (such as straight weld or circular weld), the position and attitude trajectory in the tool coordinate system are defined in the motion controller. The desired operation trajectory in Cartesian space is preset in the motion controller. This trajectory will clearly define the real-time position (such as x, y, z axis coordinates) and attitude of the welding torch in the tool coordinate system (described by Euler angles to ensure that the welding torch always fits the weld surface), and will be discretized into trajectory points according to the control cycle and stored in the trajectory buffer of the controller.

[0069] The first-order time-domain differential operation is performed on the obtained joint desired position. The difference between the joint angle at the current time and the previous time is divided by the control period to obtain the motion rate that the joint needs to achieve. In order to avoid speed jumps caused by small fluctuations in the trajectory points, the differential result is also low-pass filtered to make the desired speed smoother.

[0070] Extract the expected joint position and filtered expected velocity corresponding to the current moment from the buffer to ensure that these data are strictly aligned in time with the actual joint position and velocity acquired later, so as to provide accurate reference instructions for subsequent calculation of tracking error and execution of control algorithm;

[0071] Through the above steps, the welding operation requirements are accurately converted into angle and speed commands that the joints can execute. Whether it is a straight line, a circular arc or a complex weld trajectory, a seamless mapping from the operation path to the joint movement can be achieved. The discretized trajectory points are highly matched with the welding requirements, and the retrieved expected data is synchronized with the actual state sampling time. This provides a precise benchmark for subsequent tracking error calculation and adaptive robust control, ensuring the accuracy of weld tracking.

[0072] S13. Normalize the position tracking error, speed tracking error and load torque to form a standardized mapping parameter adjustment input vector;

[0073] By combining the rated parameters of the heavy-duty welding robot (such as the maximum tracking error of the joint and the rated torque) and the measured data of extreme working conditions (load change and impact disturbance), the actual value range of the tracking error, speed tracking error and load torque is calibrated. The three indicators are linearly normalized to eliminate the differences in dimensions and numerical ranges between them, so that all indicators are uniformly mapped to the [0,1] interval.

[0074] Outlier handling and smoothing optimization are performed. If a certain original index exceeds the preset calibration range, its normalized result is directly truncated to 0 or 1 to avoid extreme values ​​interfering with subsequent mapping parameter adjustments. The three smoothed normalized parameters are combined in the order of position tracking error normalized value x1, speed tracking error normalized value x2, and load torque normalized value x3 to form a standardized mapping parameter adjustment input vector.

[0075] In addition, S20 uses the standardized ternary input vector (position tracking error, velocity tracking error, and normalized load torque value) as the input to TS fuzzy inference. This vector comprehensively reflects the robot's motion accuracy and load state. Through the preset TS fuzzy rule library, it infers and outputs differential homeomorphic mapping parameters that are adapted to the current state, realizing dynamic parameter updates. Based on the updated mapping parameters, the actual position and velocity state in the joint space are nonlinearly mapped to the new coordinate space. The state, velocity, and Jacobian matrix corresponding to the coordinate transformation in the new coordinate space are calculated and output simultaneously. The new coordinate space can effectively decouple the joint coupling characteristics and suppress nonlinear interference, making the state and velocity signals smoother. The synchronously output Jacobian matrix directly supports the subsequent inverse mapping of the coordinate space.

[0076] The specific steps of S20 in updating the differential homeomorphism mapping parameters through TS fuzzy inference are as follows:

[0077] S21. Perform fuzzy set partitioning on the normalized ternary input vector, design membership function based on welding scenario characteristics, and construct fuzzy rule base based on welding process and experimental data.

[0078] The construction process of the TS fuzzy rule base is as follows:

[0079] Taking the adjustment of differential homeomorphism mapping parameters of heavy-duty robots as an example, the goal of the rule base is directly linked to the control performance, rather than being a vague expression. The accuracy goal is deduced from the welding process requirements and error tolerance, the decoupling goal is determined based on the dynamic coupling characteristics and accuracy correlation, and the reversibility goal is determined based on matrix stability and control safety.

[0080] The rule base takes a ternary normalized vector as input, which reflects the robot’s real-time operating conditions, and outputs a correction amount used to update the differential homeomorphism mapping parameters. There are two output variables, which are the correction amounts of mapping parameters a and b, and their range is strictly limited to the invertible parameter domain of the differential homeomorphism.

[0081] The core of the rule base is to convert continuous inputs into discrete fuzzy language (fuzzy set partitioning), and then construct rule entries for input combination and output correction based on working condition logic, covering all typical working scenarios of heavy-duty robots. Each input variable is divided into 3 fuzzy sets (small S, medium M, large L), corresponding to three states: stable working condition, fluctuating working condition, and emergency working condition, respectively. The interval and peak value of the fuzzy sets are calibrated based on experimental data. A total of 3×3×3=27 rules are generated from the 3 input variables (3 fuzzy sets each). All rules follow the engineering logic that the larger the error and the heavier the load, the larger the correction amount, and the output correction amount strictly matches the requirement of differential homeomorphism invertibility.

[0082] The rule entries are the core of the rule base and need to cover all combinations of input fuzzy sets (3×3×3=27 rules). The output correction of each rule needs to be determined through dynamic simulation and experimental quantification. All output corrections need to be calibrated through ADAMS dynamic simulation: after traversing the working conditions, find the optimal value that meets the accuracy target, decoupling target, and reversibility target to ensure the scientific nature of the rule output.

[0083] The steps for determining the rules using ADAMS dynamic simulation are as follows:

[0084] First, a robot solid model (including robotic arm, welding torch, and cable chain) is created in SolidWorks. After importing it into ADAMS, the joint moment of inertia, friction coefficient, and load range are defined. The straight / circular arc welding trajectory is set. Communication with MATLAB is established through the ADAMS / Controls module. A joint simulation closed loop is built with ADAMS state feedback, MATLAB fuzzy controller, correction output, and ADAMS parameter update. The control cycle is 1ms, and the simulation time for each working condition is 60s.

[0085] The parameters are traversed by grid scanning and local refinement. First, the entire search range is covered with a fixed step size. Then, a dense search is performed around the high-scoring parameters. The comprehensive score is calculated for each set of rule parameters, and the optimal rule entries are selected.

[0086] Based on the experimental data distribution of typical scenarios of heavy-duty welding robots (steady-state welding, slight arc disturbance, load change, strong impact), each normalized input vector is divided into 3 fuzzy subsets (small S, medium M, large L), covering the entire interval [0,1], and adjacent subsets have reasonable overlap.

[0087] To meet the real-time control requirements of welding robots, a triangular membership function is selected. Each input variable has three fuzzy subsets corresponding to three triangular functions, with the peak position closely matching the data distribution of the welding scenario. The membership function is as follows:

[0088] ;

[0089] in, denoted as membership degree, where a is the left endpoint, b is the peak point, and c is the right endpoint.

[0090] The rule design follows the core logic that the smaller the error and the lighter the load, the smoother the mapping parameters (weak nonlinear decoupling), and the larger the error and the heavier the load, the stronger the mapping parameters (strong nonlinear decoupling). Priorities are set: position tracking error x1 is greater than load torque x3, which is greater than speed tracking error x2, because position accuracy (weld tracking) is the core indicator in the welding process, followed by load coupling stability, and finally speed stability.

[0091] The fuzzy rule base takes the form of a rule where each rule uses an if-then structure, as follows:

[0092] If x1 is A, x2 is B, and x3 is C, then the parameter θ = k of the differential homeomorphism mapping.

[0093] S22. For the normalized input variables acquired in real time, calculate their membership degree to each fuzzy set according to the membership function, quantify the degree to which the input variables belong to each fuzzy state, screen out activation rules with a membership degree greater than 0, and perform a weighted average of the outputs of all activation rules to obtain the normalized correction amount.

[0094] The collected and normalized ternary input vectors are used to calculate the membership degree values ​​of each variable belonging to the three fuzzy sets of small (S), medium (M), and large (L) according to the formula of the above-mentioned triangular membership function. These values ​​are between 0 and 1. The larger the value, the higher the degree to which the input variable belongs to the corresponding fuzzy state. In this way, abstract working condition descriptions such as error magnitude and load weight are transformed into quantifiable judgment criteria.

[0095] For each rule in the fuzzy rule base, the trigger strength (i.e., activation strength) of the rule is obtained by multiplying the membership degrees of the three fuzzy subsets of the input variables corresponding to its preconditions, as shown in the following formula:

[0096] ;

[0097] in, The activation strength of the rule. Let x1 be the membership degree. Let x2 be the membership degree. Let x3 be the membership degree.

[0098] Rules with an activation degree greater than 0 are selected. These activated rules mean that the current working condition matches the rule's preconditions and are effective rules that have practical guiding significance for the current scenario. The remaining rules with an activation degree of 0 are judged to be irrelevant to the current working condition and are not included in subsequent calculations.

[0099] In order to better determine the weight of each rule, when S22 calculates the weighted average of the output of all activation rules, the activation strength of the activation rule is calculated by the product method, that is, the product of the membership degree of the input variables, which reflects the comprehensive matching degree between the input state and the rule antecedent, and the activation strength of the rule is used as the weight.

[0100] Using the activation strength of each activation rule as the weight, a weighted average is calculated for the output correction of all activation rules: first, the activation strength of each rule is multiplied by its corresponding output correction; then, all products are summed; finally, the result is divided by the sum of the activation strengths of all activation rules to obtain the final normalized correction. The formula is as follows:

[0101] ;

[0102] in, For the final correction amount, For the first The activation strength of the rule, For the first The output correction amount of the rule, The number of rules to activate.

[0103] The output results of all activated rules (i.e., the correction amount of the differential homeomorphic mapping parameters) are calculated by weighting the average value according to the proportion of each rule's activation degree. The higher the activation degree of the rule, the greater the weight of its output correction amount in the final result. In this way, the final normalized correction amount is obtained. This correction amount can be continuously and smoothly adjusted with changes in input variables without abrupt changes. It can accurately adapt to the current welding conditions and provide reliable support for the real-time update of the subsequent differential homeomorphic mapping parameters.

[0104] The activation intensity not only intuitively reflects the comprehensive matching degree between the input state and the rule antecedent, but also directly serves as the weight to dominate the calculation of the correction amount. The higher the matching degree of the rule, the greater the impact of its output correction amount on the final result, ensuring that the correction amount can accurately adapt to the current welding condition. At the same time, the weighted average calculation logic makes the correction amount change continuously with the input state, avoiding parameter abrupt changes when switching conditions, and ensuring the stability of the heavy-duty welding robot's movement.

[0105] S23. Based on the mapping parameters of the previous moment, add the amplitude-limited correction amount to obtain the mapping parameters of the current moment, ensuring that the updated parameters are within the preset range, so as to maintain the reversibility and decoupling effect of the mapping.

[0106] The baseline value of the mapping parameters at the previous moment is clearly defined. This value is a valid parameter verified by the previous control cycle to ensure that the mapping is reversible and can achieve good decoupling. It serves as the basis for the current parameter update to avoid the parameter update from being out of touch with the historical state. The normalized correction amount is calculated and subjected to amplitude limiting. Based on the process requirements of the heavy-duty welding robot and the experimental calibration results, the maximum adjustment range of the correction amount is preset. That is, when the normalized correction amount exceeds the maximum adjustment range, it is directly truncated to the boundary of the interval. This is done to avoid the correction amount being too large, which would cause parameter abrupt changes and thus destroy the continuity and reversibility of the mapping.

[0107] The reference parameters from the previous moment are added to the correction amount after limiting to obtain the initial mapping parameters for the current moment. The safe range of the mapping parameters is preset. This range is the key range to ensure the reversibility of the differential homeomorphism mapping (ensuring that the new coordinate space and the joint space can be uniquely mapped in both directions) and effective decoupling (the joints can still be separated from each other under strong coupling conditions).

[0108] To ensure that the invertibility condition is still met after the parameters of the differential homeomorphism are adaptively adjusted, strict mathematical constraints are imposed on the form of the mapping function and the parameter space. The essence of the differential homeomorphism is bidirectional smooth and invertible. For the polynomial type differential homeomorphism used in heavy-duty robots, the key to its invertibility lies in the non-singularity of the Jacobian matrix, that is, the determinant of the matrix is ​​not 0, or the minimum singular value is greater than 0.

[0109] For this polynomial mapping, the Jacobian matrix is ​​a diagonal matrix (each joint is mapped independently, with no coupling interference), and the... The diagonal elements (local mapping derivatives) corresponding to each joint are:

[0110] ;

[0111] To ensure the mapping is reversible, it is necessary to ensure that it occurs within the robot's full range of motion (e.g., joint rotation angles are typically ±180 degrees, corresponding to...). For all joints ∈ [−π,π], the Jacobian matrix is ​​not zero. Combined with engineering safety margins (to avoid critical singularities), the mathematical conditions are further transformed into parameter inequality constraints, as follows:

[0112] When the joint moves to its maximum position ;

[0113] When the joint moves to its minimum position ;

[0114] Obtain the actual motion limits of each joint from the robot technical manual or actual measurement data, and clarify the maximum and minimum positions of the joint motion. The load range of the heavy-duty robot is 300 to 600 kg. The increase of load will lead to an increase in joint inertia moment and coupling torque. It is necessary to appropriately increase parameter a to enhance the nonlinear decoupling effect. The relationship between parameter a and Jacobian stability under different loads is simulated using ADAMS dynamic simulation software to determine the load correction coefficient.

[0115] Substituting the joint motion range and load correction coefficient into the Jacobian nonsingularity inequality constraints, the basic range of parameters is obtained by solving the simultaneous equations. On a heavy-duty robot test bench, 5 to 8 sets of parameters are selected according to the mapping parameter invertibility boundary table. The robot is controlled to move along typical trajectories (straight weld seam, circular arc weld seam) in welding operations. The minimum singular value of the Jacobian matrix of each joint is collected in real time. The triggering conditions and effects of each parameter adjustment are recorded. The data is analyzed periodically (e.g., weekly). If parameter contraction is frequently triggered under a certain load, it indicates that the offline range corresponding to that load is too loose and needs to be recalibrated to reduce the range.

[0116] S24. Based on the polynomial-type differential homeomorphism mapping function, the actual position in the joint space is converted into the state in the new coordinate space, and the Jacobian matrix is ​​derived based on the partial derivative of the mapping function.

[0117] The form of the polynomial-type differential homeomorphism mapping function is determined. To balance computational efficiency and decoupling effects, a second-order univariate polynomial is chosen as the mapping relationship for a single joint (in multi-joint scenarios, each joint is mapped independently, the dimension of the new coordinate space is consistent with the number of joints, ensuring the invertibility of the differential homeomorphism). The mapping function is defined as follows:

[0118] ;

[0119] in, For the first in the new coordinate space Each state component For the first The actual position of each joint The mapping parameters at the current moment. For the number of joints, , , These are the mapping coefficients.

[0120] Second-order polynomials possess both nonlinear decoupling capabilities (which can counteract the effects of nonlinear coupling between joints) and meet the core requirements of smooth and reversible differential homeomorphism.

[0121] Substituting the actual positions of each joint into the mapping function generates the state of the new coordinate space. For each joint of the heavy-duty welding robot, the mapping function is used to calculate the complete state vector of the new coordinate space. The Jacobian matrix is ​​derived based on the partial derivatives of the mapping function. The Jacobian matrix of the coordinate transformation describes the local linear mapping relationship between the state of the new coordinate space and the position of the joint space, which is the key to the subsequent inverse mapping of the control input of the new coordinate space to the torque of the joint space. Since a single-joint independent polynomial mapping is used, the Jacobian matrix is ​​a diagonal matrix (off-diagonal elements are 0 for simplified calculation), and its element in the i-th row and i-th column is the partial derivative of the mapping function with respect to the position of the i-th joint.

[0122] ;

[0123] in, For Jacobian matrices, To beg The partial derivatives, For the first The actual position of each joint.

[0124] The Jacobian matrix of a differential homeomorphism must be nonsingular (ensuring the existence of the inverse mapping), while the partial derivatives of a second-order polynomial mapping (i.e., the diagonal elements of the Jacobian matrix) are due to... Greater than 0 and It is positive (as ensured by calibration), always greater than 0, satisfies the non-singularity requirement, and ensures the effectiveness of subsequent inverse mapping;

[0125] It achieves the decoupled transformation from joint space state to new coordinate space (the new coordinate space has no coupling interference and the state is simpler), and simultaneously obtains the non-singular Jacobian matrix, providing a precise mathematical tool for subsequent control input inverse mapping and torque calculation. Moreover, the entire transformation and derivation process is computationally efficient and fully adaptable to the real-time control requirements of heavy-duty welding robots.

[0126] Furthermore, in the new coordinate space, S30 acquires the state and velocity information of each joint's topological neighbors. By calculating the consistency error of multi-joint synchronous motion, it generates a cooperative control term to compensate for joint coupling, ensure synchronization, and intervene in control in advance as feedforward information. It integrates the position tracking error and velocity tracking error in the local new coordinate space with the feedforward information of the cooperative control term to construct a sliding surface that balances tracking accuracy and synchronization, providing an error benchmark for robust control. Combined with the updated mapping parameters after TS fuzzy inference (adapting to the decoupling and disturbance rejection requirements of the current working condition), it designs a robust control term based on the sliding surface to suppress external disturbances (such as electric arc impact) and parameter perturbations. According to the dynamic change trend of the sliding surface (such as convergence rate) and the consistency error of the cooperative control term, it estimates uncertainties (such as load mutation and joint wear) online through an adaptive law to generate an adaptive compensation term to accurately offset unknown interference. The cooperative control term, robust control term, and adaptive compensation term run in parallel to form a comprehensive control output in the new coordinate space, laying the foundation for subsequent inverse mapping and torque fusion.

[0127] The specific steps for S30 to calculate the cooperative control term, robust control term, and adaptive compensation term are as follows:

[0128] S31. Define the joint neighbor relationship through distributed communication topology, calculate the state consistency error between this joint and neighbor joints, and combine it with the expected trajectory tracking requirements to form the total cooperative deviation. Synthesize the cooperative control term through the proportional-derivative control structure.

[0129] Based on the mechanical structure and control requirements of heavy-duty welding robots, a fixed communication topology is adopted, and the topological neighbors of each joint are clearly defined. In the new coordinate space, the real-time state zi(k) of the current joint and the real-time states zj(k) of the neighboring joints (j is the sequence number of the neighboring joint) are obtained, and a consistency error is defined. This is used to quantify the synchronization deviation between this joint and neighboring joints. The closer the consistency error is to 0, the better the synchronization between the two joints.

[0130] By synthesizing a cooperative control term using a proportional-derivative (PD) control structure, and to quickly correct the total cooperative deviation and suppress abrupt deviation changes, a PD control law is used to design the cooperative control term, the expression of which is:

[0131] ;

[0132] in, For collaborative control items, This is a scaling factor (based on experimental calibration), used to amplify the current total deviation and accelerate convergence. These are differential coefficients (based on experimental calibration) used to suppress the rate of change of deviation and avoid overshoot and chattering. For consistency error, Consistency error The differential.

[0133] The synthesized cooperative control term has a dual function: first, by correcting the consistency error, it aligns the motion state of the joint with that of neighboring joints, ensuring the synchronization of multiple joints; second, by incorporating local tracking error, it ensures that the synchronous motion does not deviate from the desired trajectory, providing reliable feedforward information for subsequent sliding surface construction and robust control.

[0134] S32. Calculate the tracking error in the new coordinate space, including position error and velocity error, and construct a linear sliding surface. Combine the mapping parameters output by TS fuzzy inference to design a boundary layer thickness with adaptive characteristics and generate a robust control term.

[0135] Position tracking error: The deviation between the actual state and the desired state of this joint in the new coordinate space is defined as the position error. The desired state is obtained by inverse solution and mapping transformation of the desired trajectory in Cartesian space, which directly reflects the trajectory tracking deviation in the new coordinate space.

[0136] Speed ​​tracking error: Calculated directly based on the actual speed and the expected speed in the new coordinate space, reflecting the dynamic deviation of speed tracking;

[0137] To construct a linear sliding surface and achieve rapid error convergence, a linear sliding surface structure weighted by position and velocity errors is adopted, expressed as follows:

[0138] ;

[0139] in, In the state of sliding surface, The position tracking error under the new coordinates, For the velocity tracking error under the new coordinates, These are positive definite coefficients.

[0140] A boundary layer is set near the sliding surface. S32 uses a saturation function instead of a sign function to divide the range of sliding surface error into two regions: inside and outside the boundary layer. Response characteristics are designed for each region. Outside the boundary layer, the saturation function maintains a step output consistent with the sign function. Inside the boundary layer, the saturation function no longer has a step output, but a smooth linear output.

[0141] With sliding surface With this as the core, combined with adaptive boundary layer thickness (Dynamically adjusted by TS fuzzy mapping parameters) the sliding surface error is divided into two regions, inside the boundary layer and outside the boundary layer, and the output characteristics of the saturation function change with the region switching;

[0142] Outside the boundary layer: The saturation function maintains a step output consistent with the sign function. At this time, the robust control term outputs with a constant maximum gain, which can quickly cancel large disturbances, ensure the sliding surface converges quickly, and maintain the disturbance rejection unchanged. Inside the boundary layer: The saturation function is no longer a step output, but a smooth linear output. The output value changes linearly with the sliding surface error (range [-1,1]). The corresponding robust control term output also shows a linear gradual change, avoiding the step jump of the sign function when the error approaches 0, completely suppressing chattering, and ensuring smooth joint movement.

[0143] The saturation function is: ;

[0144] The step output characteristics are retained outside the boundary layer, and the ability to suppress large disturbances is consistent with that of traditional sign functions. This ensures that disturbances such as arc impact and sudden load changes during welding can be quickly canceled out. The linear output inside the boundary layer eliminates the step jump when the error approaches 0. The joint torque output has no high-frequency fluctuations, and the motion stability is significantly improved, avoiding weld displacement due to chattering. The response characteristics of the saturation function are adjusted synchronously with the dynamic boundary layer, taking into account both the anti-disturbance requirements of complex working conditions and the stability requirements of steady-state welding, without the need for manual parameter intervention.

[0145] In order to dynamically adjust the robust control term gain, S32 constructs a comprehensive coordination state evaluation index based on the output amplitude index and the rate of change index of the coordinated control term, divides the coordination state into four levels according to the numerical range of the comprehensive coordination state evaluation index, establishes a corresponding fourth-order gain adjustment strategy, and implements the fourth-order gain adjustment strategy based on the index to dynamically adjust the robust control term gain.

[0146] From the collaborative control terms generated in step S31, two key evaluation indicators are extracted: one is the output amplitude indicator, which is the absolute value of the collaborative control term, reflecting the correction intensity of the current multi-joint synchronization deviation or trajectory tracking deviation. The larger the value, the more obvious the deviation. The other is the rate of change indicator, which is calculated by dividing the difference in output amplitude within adjacent control cycles by the control cycle, reflecting the dynamic fluctuation degree of the collaborative control term. The larger the value, the more severe the change in operating conditions or disturbance.

[0147] These two indicators are normalized and mapped to the [0,1] interval to eliminate the difference in numerical range. Then, according to the welding scenario requirement that the current correction intensity takes precedence over the rate of change of the working condition, the output amplitude normalization value is assigned a weight of 0.6 and the change rate normalization value is assigned a weight of 0.4. After weighted summation, a comprehensive coordination state evaluation index (numerical range [0,1]) is obtained. This index can comprehensively quantify the coordination state of multiple joints.

[0148] Based on the numerical range of the comprehensive coordination status evaluation index, the coordination status is divided into four levels: Level 1 (Excellent) is defined as the index range of [0, 0.25], corresponding to scenarios with good synchronization of multiple joints and stable operating conditions; Level 2 (Good) is defined as the index range of (0.25, 0.5], corresponding to scenarios with slight synchronization deviation and basically stable operating conditions; Level 3 (Medium) is defined as the index range of (0.5, 0.75], corresponding to scenarios with moderate synchronization deviation and rapid changes in operating conditions; and Level 4 (Poor) is defined as the index range of (0.75, 1.0], corresponding to scenarios with severe synchronization deviation and sudden changes in operating conditions.

[0149] Based on actual working data of heavy-duty robots, four threshold standards for switching levels are set to match control requirements and take into account engineering practicality.

[0150] The threshold of 0.25 is the dividing line between stable operating conditions (Level 1) and slight fluctuations (Level 2). The core basis is the data analysis of steady-state light load scenarios, ensuring that the excellent level only covers truly undisturbed and highly synchronized operating conditions. The comprehensive coordination state evaluation index is calculated by weighting the output amplitude and rate of change of the collaborative control item (amplitude weight 0.6, rate of change weight 0.4). Its value directly corresponds to the stability of the operating conditions. For steady-state light load welding scenarios, 1000 sets of actual operating data were collected, and the distribution of the comprehensive index was statistically analyzed. 90% of the data are concentrated in the range of [0, 0.25]. At this time, the output amplitude of the collaborative control item is less than 0.3, and the rate of change is less than 0.2, which means that the synchronization deviation of the multi-joint is less than 5, the operating conditions are stable, and no strong anti-disturbance is required.

[0151] If the threshold is set to 0.3 (instead of 0.25), 20% of the slightly fluctuating data will be misclassified as excellent. At this time, using the small gain of the first stage (8.0 to 9.0) will cause the synchronization deviation to increase to 0.2mm due to insufficient disturbance rejection. If it is set to 0.2, 15% of the steady-state data will be misclassified as good, triggering the medium-to-small gain of the second stage (8.8 to 10.0), resulting in energy waste and affecting stability.

[0152] Similarly, the data distribution matching of the transitional scenario and the data analysis of the heavy-load and strong-impact scenario were used to determine the thresholds of 0.5 and 0.75 using the same method.

[0153] The classification is based on actual operating data of typical heavy-duty welding robot operation scenarios (steady-state light-load welding, slight arc disturbance, medium-load intermittent disturbance, and heavy-load strong impact). The amplitude and rate of change of the collaborative control items in each scenario are statistically analyzed. 90% of the steady-state light-load data corresponds to a comprehensive index of less than 0.25, and 85% of the heavy-load strong impact data corresponds to a comprehensive index of greater than 0.75. The intermediate range is naturally matched with transitional working conditions to ensure that the classification is strongly bound to the actual working conditions.

[0154] For the four coordination state levels, a four-order robust control gain adjustment strategy is formulated: Level 1 (excellent) uses a small gain of [8.0, 9.0] to prioritize motion stability; Level 2 (good) uses a medium-to-small gain of [8.8, 10.0] to balance stability and disturbance rejection; Level 3 (medium) uses a medium-to-large gain of [9.8, 11.0] to enhance disturbance rejection; and Level 4 (poor) uses a large gain of [10.8, 12.0] to prioritize the rapid cancellation of deviations and disturbances. To avoid abrupt gain changes during level switching, a first-order low-pass filter is applied to the adjusted gain to ensure a smooth gain transition.

[0155] The smoothed dynamic gain, combined with the boundary layer thickness adaptively adjusted by the TS fuzzy mapping parameters and the saturation function of the partitioned response, is substituted into the robust control term expression as follows:

[0156] ;

[0157] in, For the first Joint Robust control output at any time. The dynamic robust gain after fourth-order strategy adjustment and low-pass filtering. It is a saturation function. In the state of sliding surface, For adaptive boundary layer thickness.

[0158] The final robust control term is generated, which can accurately match the anti-disturbance and stability requirements under different coordination states through end-to-end adaptation, and is fully adapted to the complex working conditions of heavy-duty welding robots.

[0159] S33. The parameter uncertainty is linearized and processed. Based on the changing trend of the sliding surface and the consistency error of the cooperative control term, the parameter estimate is updated online through the adaptive law to generate the adaptive compensation term.

[0160] For the perturbations (i.e., parameter uncertainties) of parameters such as joint moment of inertia, viscous damping coefficient, and load stiffness of heavy-duty welding robots, linear parameterization is performed to transform these nonlinear uncertainties into the form of a known regression matrix × unknown constant parameters. The known regression matrix consists of the state, velocity, and consistency error of the cooperative control terms in the new coordinate space, and can be calculated and obtained in real time. The unknown constant parameter vector contains the true values ​​of each uncertain parameter, and its magnitude does not change with time. It can be offset by online estimation.

[0161] The core design basis is the changing trend of the sliding surface and the consistency error of the cooperative control term: the magnitude and rate of change of the sliding surface can reflect the intensity of the interference of uncertainty on the tracking control. The larger the value and the more violent the fluctuation, the stronger the interference. The consistency error of the cooperative control term can reflect the degree of damage of parameter perturbation to the synchronization of multiple joints. The larger the error, the more serious the impact on synchronization. Based on these two indicators, a proportional adaptive law is adopted. Based on the parameter estimate of the previous moment, the update speed is adjusted through adaptive gain to update the estimate of unknown parameters in real time, ensuring that the parameter estimate is always adjusted in the direction of offsetting uncertainty. At the same time, boundary constraints are applied to the estimate to avoid its divergence.

[0162] Substituting the online updated parameter estimates into the linear parameterized expression yields an estimate of the uncertainty. This estimate is then inverted to generate the adaptive compensation term, as shown in the formula:

[0163] ;

[0164] in, For adaptive compensation terms, For the regression matrix, These are estimated values ​​for the location parameters.

[0165] This compensation term is opposite to the direction of the uncertainty of the actual parameters, and can accurately offset the interference caused by sudden load changes, joint wear, etc. It not only ensures the accuracy of single joint trajectory tracking, but also maintains the synchronization of multiple joints. It complements the cooperative control term and the robust control term, and further enhances the robustness and adaptability of the control of heavy-duty welding robots.

[0166] The collaborative control terms effectively suppress multi-joint coupling deviations and load abrupt changes through topological neighbor information interaction, avoiding weld seam offset. The robust control terms and mapping parameters are dynamically adapted, and combined with the online estimation of the adaptive compensation terms, the suppression efficiency of unknown disturbances such as arc disturbances and load fluctuations is improved. The sliding surface integrates feedforward information and dual errors, and with the parallel output of three control terms, the adaptive law and fuzzy mapping parameters are linked, enabling the control terms to adapt to all working conditions such as steady-state welding, strong impact, and parameter drift in real time without the need for manual parameter adjustment, making it highly practical for engineering applications.

[0167] In addition, the S40 uses a Nash game equilibrium strategy to assign dynamic weights to the cooperative control term, robust sliding mode control term, and adaptive compensation term, and obtains the final control input in the new coordinate space by weighted summation. With the help of the Jacobian matrix, this input is inversely mapped to the basic control torque in the joint space. The nominal dynamic feedforward compensation term is superimposed to generate a comprehensive control torque and output to the joint actuator. It dynamically adapts to the working conditions, achieves optimal coordination of synchronization, disturbance rejection, and uncertainty compensation, improves welding tracking accuracy and stability, and optimizes actuator response efficiency.

[0168] The specific steps for S40 to generate the final integrated control torque are as follows:

[0169] S41. Treat the cooperative control term, robust sliding mode control term, and adaptive compensation term as three game participants that cooperate and constrain each other. Design a unique cost function for each participant. Through the constraint optimization conditions, derive the dynamic adjustment rules of each weight, and finally obtain the optimal weight combination that satisfies the equilibrium conditions.

[0170] The collaborative control term, robust sliding mode control term, and adaptive compensation term are treated as game participants. Each cost function is bound to its own core function. The cost function of the collaborative control term is associated with the multi-joint synchronization deviation (the larger the deviation, the higher the cost). The cost function of the robust sliding mode control term is associated with the disturbance suppression error (the worse the suppression effect, the higher the cost). The cost function of the adaptive compensation term is associated with the parameter estimation error (the more inaccurate the estimation, the higher the cost). The performance cost of each control term is quantified.

[0171] The cost function of the collaborative control term is:

[0172] ;

[0173] in, The cost function of the collaborative control term, The weighting coefficients for the consistency error cost. For the consistency error vector, The weighting coefficients for tracking error costs, For the desired trajectory tracking error, The actual tracking error after compensation by the coordinated control term. The weighting coefficients for the control quantity regularization term, This is the output value of the coordinated control term.

[0174] The relative magnitude of the weighting coefficients is determined by the priority of the control objective. It is necessary to first clarify the core requirements of the heavy-duty robot in different scenarios to avoid control imbalance caused by uniform distribution of coefficients. The core objective of the coordination term is that weld seam accuracy (tracking error) is greater than multi-joint synchronization (consistency error) and control quantity energy saving (regularization term). Therefore, the weighting coefficients satisfy α2 (tracking error weight) greater than α1 (consistency error weight) greater than α3 (regularization term weight). The core value of heavy-duty welding is weld seam quality. If the tracking error exceeds 0.8mm, it will directly lead to scrapping, while the synchronization error (such as 0.1mm) can be corrected through subsequent compensation. The regularization term is only used to avoid excessive control quantity (such as motor overload) and has the lowest priority.

[0175] The cost function of the robust control term is:

[0176] ;

[0177] in, The cost function of the robust control term. This is a weighting coefficient for the cost of sliding surface error. For the actual sliding surface, For the desired sliding surface, Weighting coefficients for the boundary layer adaptation error cost. This represents the actual boundary layer thickness. To adapt to the desired boundary layer thickness under current operating conditions, The weighting coefficients for the control quantity regularization term, This is the output value of the robust control term.

[0178] The core objective of the robust term is that the disturbance rejection effect (sliding surface error) is greater than the chattering suppression (boundary layer error) and the energy saving of the control quantity (regularization term). Therefore, the weighting coefficients satisfy β1 (sliding surface error weight) greater than β2 (boundary layer error weight) greater than β3 (regularization term weight). External disturbances (such as impact loads) will directly destroy trajectory tracking, and the convergence of the sliding surface to 0 is the key to disturbance rejection. Boundary layer error only affects motion stability (chattering) and will not directly cause weld failure, so its priority is secondary.

[0179] The cost function of the adaptive compensation term is:

[0180] ;

[0181] in, The cost function of the adaptive compensation term, The weighting coefficients for the cost of parameter estimation error. These are parameter estimates. For the actual value of the parameter, The weighting coefficients for the cost of adapting the sliding surface change rate. This represents the actual rate of change of the sliding surface. To be the desired rate of change of the sliding surface, The weighting coefficients for the control quantity regularization term, This is the output of the adaptive compensation term.

[0182] The core objective of the adaptive term is that the parameter estimation accuracy (estimation error) is greater than the compensation timeliness (sliding surface change rate) and the control quantity energy saving (regularization term). Therefore, the coefficients satisfy γ1 (parameter estimation error weight) greater than γ2 (sliding surface change rate weight) greater than γ3 (regularization term weight). Parameter uncertainty (such as joint wear) will affect the control accuracy in the long term, and inaccurate estimation will lead to compensation failure. Change rate adaptation only affects the compensation speed and has a lower priority than estimation accuracy.

[0183] Taking collaborative control terms (α1, α2, α3) and robust control terms (β1, β2, β3) as examples, the logic for determining the weighting coefficients is as follows:

[0184] First, fix the regularization coefficients (e.g., α3=0.2, β3=0.2). These coefficients only constrain the magnitude of the control quantity and have the lowest priority. Initially, set them to fixed small values. For the coordination term, fix α1 (synchronization error weight) and adjust α2 (tracking error weight). Record the tracking errors corresponding to different α2 values: Under steady-state light load conditions, when α2=0.4, the tracking error is 0.6mm (not up to standard); when α2=0.5, the error is 0.45mm (up to standard); when α2=0.6, the error is 0.4mm but the synchronization error exceeds 0.1mm (not up to standard). Finally, determine α1=0.3 and α2=0.5.

[0185] For the robustness term, fix β2 (boundary layer error weight) and adjust β1 (sliding surface error weight): under heavy load and strong impact conditions, when β1=0.6, the error recovery time after disturbance is 1.5ms (not up to standard), and when β1=0.7, the recovery time is only 0.8ms (up to standard). Therefore, determine β1=0.7 and β2=0.1.

[0186] Based on the constraints, a Lagrangian function is constructed, and the Nash equilibrium is solved by minimizing the total cost. The weight adjustment rule is derived: the smaller the error (the better the performance) of a certain control term, the larger its dynamic weight; the larger the error (the worse the performance), the smaller the weight, thus achieving dynamic adaptation of high weight for the better and low weight for the worse.

[0187] The weights of the three control items are updated in real time according to the adjustment rules, and the optimal combination that satisfies the equilibrium condition is finally obtained, which allows each control item to play its full role, avoids functional conflicts, and achieves the best overall control performance.

[0188] S42. Based on the dynamic weights obtained from the Nash equilibrium solution, the three control terms are weighted and summed to obtain the comprehensive control input of the new coordinate space. Using the bridge established by the Jacobian matrix, the comprehensive control input of the new coordinate space is inversely mapped to the original joint space to obtain the basic control torque of the joint space.

[0189] Based on the dynamic weights obtained from the Nash equilibrium solution, the cooperative control term, robust sliding mode control term, and adaptive compensation term are weighted and summed. The weights directly match the performance priority of each control term under the current operating condition, ensuring that synchronization guarantee, disturbance suppression, and uncertainty compensation functions work together to ultimately obtain the comprehensive control input in the new coordinate space, achieving the optimal integration of control requirements within this space.

[0190] Using the non-singular Jacobian matrix derived in step S24 as the core bridge, and leveraging the mapping relationship between the new coordinate space and the original joint space established by it, the comprehensive control input of the new coordinate space is accurately converted into the basic control torque of the original joint space through inverse mapping operation (based on the duality requirement of torque mapping, using Jacobian matrix transpose). This conversion process strictly follows the invertibility of differential homeomorphism mapping, ensuring that the optimal control logic of the new coordinate space can be accurately transmitted to the joint execution level, providing a reliable foundation for subsequent torque optimization.

[0191] Considering the case where the Jacobian matrix is ​​singular or nearly singular, S42 employs a least-squares pseudo-inverse algorithm with a damping factor to solve the inverse transformation when the Jacobian matrix is ​​singular. The damping factor is dynamically adjusted based on the singular value spectrum of the Jacobian matrix. By calculating the condition number and the minimum singular value of the matrix in real time, a nonlinear mapping relationship between the damping factor and the ill-conditioned degree of the matrix is ​​established, minimizing the joint moment norm and limiting the spatial gain of the solution.

[0192] The dynamic weights derived from Nash equilibrium are used to sum the three control terms to obtain the new coordinate space integrated control input. Next, it is determined whether the Jacobian matrix is ​​singular by calculating the minimum singular value and the condition number. If it is singular or nearly singular, the least squares pseudo-inverse with dynamic damping is used. The damping factor is nonlinearly adjusted according to the ill-conditioned nature of the matrix (the smaller the singular value and the larger the condition number, the greater the damping). Finally, the pseudo-inverse formula is used for singular scenarios, and the matrix transpose is used for non-singular scenarios to inversely map the integrated control input into the basic control torque in the joint space, ensuring that the torque is stable and does not diverge.

[0193] S43. Based on the basic control torque, a feedforward compensation term based on the nominal dynamics model is superimposed to form a comprehensive control torque;

[0194] First, based on the robot's nominal dynamics model (including known joint inertia moments, Coriolis / centrifugal forces, gravity terms, etc.), a feedforward compensation term is calculated to offset the foreseeable dynamic coupling and gravity effects during joint movement in advance. Then, this feedforward compensation term is superimposed with the joint space basic control torque obtained by S42 to finally form a comprehensive control torque that takes into account both basic adjustment and disturbance pre-cancellation, thereby reducing the burden of basic torque adjustment and improving control response accuracy.

[0195] In summary, the working principle of this solution is as follows:

[0196] This adaptive robust control method for a heavy-duty robot, which integrates differential homeomorphism mapping, collects real-time information on the actual position, velocity, and welding torch load torque of each joint and compares it precisely with the desired trajectory. It outputs a three-dimensional input vector containing position tracking error, velocity tracking error, and load torque, establishing a precise foundation for working condition identification and providing a reliable input basis for subsequent intelligent control. Through real-time error calculation, it ensures the system's rapid response capability to dynamic changes.

[0197] Based on a ternary input vector, the differential homeomorphism mapping parameters are updated in real time through the TS fuzzy inference system. Based on the optimized parameters, the joint space state is mapped to a new coordinate space, and the state, velocity, and Jacobian matrix of the new coordinate space are output synchronously. Through adaptive adjustment of the mapping parameters, the system can automatically optimize the transformation characteristics according to different operating conditions. The differential homeomorphism mapping effectively decouples the nonlinear characteristics of the system and greatly simplifies the subsequent control design. The real-time calculation of the Jacobian matrix provides a precise mathematical basis for the inverse transformation of the control force.

[0198] Within the new coordinate space, the system acquires neighbor joint information through distributed communication to generate cooperative control terms, constructs a sliding surface based on local tracking errors to generate robust control terms, and generates compensation terms based on adaptive law to estimate parameter uncertainties, forming three parallel control outputs. Distributed cooperative control ensures the synchronization and coordination of multi-joint motion; robust sliding mode control provides strong anti-disturbance capability; adaptive compensation realizes online learning and cancellation of parameter uncertainties; the parallel generation and coordinated linkage of the three controls form a complete control and protection system.

[0199] A Nash game equilibrium strategy is adopted to dynamically allocate and fuse the three control terms. The basic control torque is obtained through the inverse transformation of the Jacobian matrix, and then a feedforward compensation term based on the nominal dynamics model is superimposed to generate the final comprehensive control torque. The Nash game equilibrium realizes intelligent trade-offs and optimization of multiple control objectives. The damping factor design of the pseudo-inverse algorithm ensures the stability of the control in the singular region. The dynamic feedforward compensation improves the system's response speed and tracking accuracy. The final comprehensive control torque optimizes energy distribution while ensuring performance.

[0200] The foregoing has shown and described the basic principles, main features, and advantages of the present invention. Those skilled in the art should understand that the present invention is not limited to the above embodiments. The embodiments and descriptions in the specification are merely preferred examples and are not intended to limit the invention. Various changes and modifications can be made to the invention without departing from its spirit and scope, and all such changes and modifications fall within the scope of the present invention as claimed. The scope of protection of the present invention is defined by the appended claims and their equivalents.

Claims

1. An adaptive robust control method for a heavy-duty robot incorporating differential homeomorphism mapping, characterized in that, Includes the following steps: S10. Collect the actual position, speed and welding torch load torque information of each joint of the heavy-duty welding robot, compare it with the expected trajectory and output a three-dimensional input vector composed of position tracking error, speed tracking error and load torque. S20. Using the ternary input vector as input, update the differential homeomorphism mapping parameters through TS fuzzy inference. Based on these parameters, map the joint space state to the new coordinate space, and simultaneously output the state, velocity, and Jacobian matrix of the new coordinate space. The mapping process is as follows: The joint position is transformed into a new coordinate space position through the decoupling matrix of the kernel function and the associated mapping parameters. The joint velocity, combined with the rate of change of the mapping parameters and the coupling relationship between position and velocity, is transformed into a new coordinate space velocity through the combined action of the kernel function derivative and the kernel function. S30. In the new coordinate space, establish a distributed collaborative, robust control, and adaptive compensation linkage generation mechanism: The cooperative control term is calculated based on the state information of neighboring joints, and its output is used as a feedforward signal. A sliding surface is constructed in combination with the local tracking error. The boundary layer is adaptively adjusted according to the mapping parameters output by fuzzy inference to generate a robust control term. At the same time, based on the changing trend of the sliding surface and the cooperative consistency error, a compensation term is estimated online through an adaptive law to generate a total of three parallel and cooperative control outputs. S40. A Nash game equilibrium strategy is adopted to perform multi-objective dynamic weight allocation and fusion of the three control terms. The basic control torque is obtained through Jacobi matrix inverse mapping. Then, combined with feedforward dynamic compensation, the final comprehensive control torque is generated and output to each joint actuator.

2. The adaptive robust control method for a heavy-load robot based on fusion of differential homeomorphisms as described in claim 1, characterized in that: The steps in S10 for calculating the position tracking error and velocity tracking error of each joint of the heavy-duty welding robot are as follows: S11. Real-time acquisition of joint rotation angle signals, decoding by the controller to obtain the actual position of each joint, differential operation on the continuously sampled actual position signals to obtain the actual speed of each joint, and connecting a piezoelectric torque sensor in series at the drive end of each joint to acquire the joint output torque in real time. S12. Retrieve the preset joint expected trajectory data from the motion controller of the welding robot, extract the expected position and expected speed of the joint at each moment, compare the actual position and actual speed of each joint with the expected position and expected speed of the joint at each moment, and obtain the position tracking error and speed tracking error of each joint of the heavy-duty welding robot. S13. Normalize the position tracking error, speed tracking error, and load torque to form a standardized mapping parameter adjustment input vector.

3. The adaptive robust control method for a heavy-load robot based on fusion of differential homeomorphisms as described in claim 2, characterized in that: S12, based on the operational requirements of the heavy-duty welding robot, presets the desired operation trajectory in Cartesian space in the robot motion controller, uses the robot forward kinematics inverse algorithm to convert the desired operation trajectory in Cartesian space into the angular space trajectory of each joint, obtains the desired joint position at each time, performs first-order differential operation in the time domain on the desired joint position to obtain the desired joint velocity, and retrieves the preset desired joint trajectory data.

4. The adaptive robust control method for a heavy-load robot based on fusion differential homeomorphism mapping as described in claim 3, characterized in that: The specific steps of S20 in updating the differential homeomorphism mapping parameters through TS fuzzy inference are as follows: S21. Perform fuzzy set partitioning on the normalized ternary input vector, design membership function based on welding scenario characteristics, and construct fuzzy rule base based on welding process and experimental data. S22. For the normalized input variables acquired in real time, calculate their membership degree to each fuzzy set according to the membership function, quantify the degree to which the input variables belong to each fuzzy state, screen out activation rules with a membership degree greater than 0, and perform a weighted average of the outputs of all activation rules to obtain the normalized correction amount. S23. Based on the mapping parameters of the previous moment, add the amplitude-limited correction amount to obtain the mapping parameters of the current moment, ensuring that the updated parameters are within the preset range, so as to maintain the reversibility and decoupling effect of the mapping. S24. Based on the polynomial-type differential homeomorphism mapping function, the actual position in the joint space is converted into the state in the new coordinate space, and the Jacobian matrix is ​​derived based on the partial derivative of the mapping function.

5. The adaptive robust control method for a heavy-load robot based on fused differential homeomorphism mapping according to claim 4, characterized in that: When S22 performs a weighted average of the outputs of all activation rules, the activation strength of the activation rule is calculated by the product method, which is the product of the membership degrees of the input variables. This reflects the comprehensive matching degree between the input state and the rule's antecedents, and the activation strength of the rule is used as the weight.

6. The adaptive robust control method for a heavy-load robot based on fusion differential homeomorphism mapping as described in claim 1, characterized in that: The specific steps for S30 to calculate the cooperative control term, robust control term, and adaptive compensation term are as follows: S31. Define the joint neighbor relationship through distributed communication topology, calculate the state consistency error between this joint and neighbor joints, and combine it with the expected trajectory tracking requirements to form the total cooperative deviation. Synthesize the cooperative control term through the proportional-derivative control structure. S32. Calculate the tracking error in the new coordinate space, including position error and velocity error, and construct a linear sliding surface. Combine the mapping parameters output by TS fuzzy inference to design a boundary layer thickness with adaptive characteristics and generate a robust control term. S33. The parameter uncertainty is linearized and processed. Based on the changing trend of the sliding surface and the consistency error of the cooperative control term, the parameter estimate is updated online through the adaptive law to generate the adaptive compensation term.

7. The adaptive robust control method for a heavy-load robot based on fusion of differential homeomorphisms as described in claim 6, characterized in that: The S32 uses a saturation function instead of a sign function to divide the range of sliding surface error into two regions: inside and outside the boundary layer. Response characteristics are designed for each region. Outside the boundary layer, the saturation function maintains a step output consistent with the sign function. Inside the boundary layer, the saturation function no longer provides a step output but instead provides a smooth linear output.

8. The adaptive robust control method for a heavy-load robot based on fused differential homeomorphism mapping as described in claim 6, characterized in that: S32 constructs a comprehensive coordination state evaluation index based on the output amplitude index and the rate of change index of the coordinated control term. According to the numerical range of the comprehensive coordination state evaluation index, the coordination state is divided into four levels, and a corresponding fourth-order gain adjustment strategy is established. Based on the index, the fourth-order gain adjustment strategy is implemented to dynamically adjust the gain of the robust control term.

9. The adaptive robust control method for a heavy-load robot based on fusion of differential homeomorphisms according to claim 1, characterized in that: The specific steps for S40 to generate the final integrated control torque are as follows: S41. Treat the cooperative control term, robust sliding mode control term, and adaptive compensation term as three game participants that cooperate and constrain each other. Design a unique cost function for each participant. Through the constraint optimization conditions, derive the dynamic adjustment rules of each weight, and finally obtain the optimal weight combination that satisfies the equilibrium conditions. S42. Based on the dynamic weights obtained from the Nash equilibrium solution, the three control terms are weighted and summed to obtain the comprehensive control input of the new coordinate space. Using the bridge established by the Jacobian matrix, the comprehensive control input of the new coordinate space is inversely mapped to the original joint space to obtain the basic control torque of the joint space. S43. Based on the basic control torque, a feedforward compensation term based on the nominal dynamics model is superimposed to form a comprehensive control torque.

10. The adaptive robust control method for a heavy-load robot based on fused differential homeomorphism mapping according to claim 9, characterized in that: When the Jacobian matrix is ​​singular, S42 uses a least-squares pseudo-inverse algorithm with a damping factor to perform an inverse transformation solution. The damping factor is dynamically adjusted based on the singular value spectrum of the Jacobian matrix. By calculating the condition number and the minimum singular value of the matrix in real time, a nonlinear mapping relationship between the damping factor and the ill-conditioned degree of the matrix is ​​established, minimizing the joint moment norm and limiting the spatial gain of the solution.

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