Intelligent spraying robot multi-task cooperative operation method and system

Abstract

The application provides a kind of intelligent spraying robot multi-task collaborative work method and system, comprising: based on discrete state equation, a discrete sliding mode controller considering sampling period is constructed;Adaptive barrier function is designed by introducing auxiliary state variable;A T-product dynamic system control model is established to realize online parameter identification;Multi-robot collaborative control is realized by using hierarchical control architecture.This scheme solves the problems of precision reduction in discrete implementation, difficulty in adaptive adjustment of controller parameters and lack of effective data-driven control strategy in the prior art, and realizes efficient collaborative work of multiple spraying robots.

Description

Technical Field

[0001] This application relates to the field of industrial automation technology, and in particular to an intelligent spraying robot multi-task collaborative operation method and system. Background Technology

[0002] Intelligent painting robots are an important research direction in the field of industrial automation, mainly applied in scenarios requiring large-area, high-precision painting, such as automobile manufacturing and furniture processing. Multi-robot collaborative operation has become a key technology for improving production efficiency.

[0003] Currently, most common painting robot systems employ either single-machine operation or simple multi-machine collaborative modes. For example, they achieve fixed-point painting by multiple robots through pre-defined trajectory planning, or use basic obstacle avoidance algorithms for simple cooperative motion control. However, these systems often suffer from low collaborative efficiency and insufficiently precise trajectory planning when handling complex workpieces or high-efficiency painting tasks.

[0004] More advanced technologies employ a robot trajectory tracking method based on sliding mode control, which ensures system stability by constructing a Lyapunov function. This method introduces nonlinear terms into the controller design to compensate for system uncertainties and external disturbances, thereby improving the system's robustness and control accuracy to a certain extent.

[0005] However, existing technologies still have the following problems: traditional sliding mode controllers often ignore the influence of the sampling period when implemented discretely, resulting in a decrease in control accuracy; controller parameters are difficult to adaptively adjust according to actual working conditions, resulting in poor system adaptability; there is a lack of effective data-driven control strategies for complex dynamic systems, making it difficult to accurately describe the dynamic characteristics of the system; and multi-robot collaborative control strategies are imperfect, making it difficult to achieve efficient collaborative operations. Summary of the Invention

[0006] In view of this, this application provides an intelligent spraying robot multi-task collaborative operation method and system, which solves the problems in the prior art where the traditional sliding mode controller ignores the influence of the sampling period when it is implemented discretely, resulting in a decrease in control accuracy, the controller parameters are difficult to adaptively adjust according to the actual working conditions, and there is a lack of effective data-driven control strategies for complex dynamic systems.

[0007] This application provides an intelligent spraying robot multi-task collaborative operation method, including:

[0008] The joint angles, angular velocities, torque data, and control parameters of the painting robot are collected as state data. Based on the state data, a discrete state equation containing the sampling period T is established. A sampling period compensation term is introduced to design a sliding mode switching surface, a discrete Lyapunov function is constructed, and the first control quantity is generated.

[0009] Based on the first control variable, combined with robot joint angle constraints, speed constraints, acceleration constraints, and workspace boundary constraints, an adaptive barrier function is constructed, and the barrier function parameters are optimized using the gradient descent method to generate the second control variable.

[0010] Based on the second control variable, the input and output data of the spraying robot during online operation are collected to construct a T-product dynamic system control model, realize online identification and updating of model parameters, and generate a third control variable;

[0011] Based on the third control variable, a multi-robot kinematics and dynamics model is established, and a hierarchical control strategy is adopted to realize multi-robot collaborative control. The output control commands meet the requirements of workload balance and spraying quality, and control multiple spraying robots to complete collaborative operations.

[0012] Based on the state data, a discrete state equation containing a sampling period T is established, including:

[0013] Taylor expansion of the robot dynamics equations at the sampling time is performed, and second-order terms are retained to establish discrete state equations containing the sampling period T.

[0014] Based on the discrete state equations, an improved sliding mode switching surface is designed, and a quadratic discrete Lyapunov function is constructed.

[0015] By analyzing the forward difference of the discrete Lyapunov function, the stability condition is obtained;

[0016] Based on the stability condition, a discrete sliding mode controller containing an equivalent control term and a switching control term is designed to generate the first control quantity.

[0017] Design a discrete sliding mode controller that includes an equivalent control term and a switching control term, including:

[0018] Based on the stability conditions, an equivalent control term is designed to compensate for the known dynamics of the system, and a switching control term is designed to counteract external disturbances.

[0019] The discrete sliding mode controller employs a continuous reaching law, setting the boundary layer thickness to a value within the range of 0.01-0.1.

[0020] The first control quantity is generated by replacing the sign function with a saturation function in the switching control item.

[0021] Constructing an adaptive barrier function includes:

[0022] Set the robot joint angle limit to ±170 degrees, the speed limit to ±180 degrees / second, and the acceleration limit to ±900 degrees / square second;

[0023] Introduce an auxiliary state variable ξ with a value range of 0.1-10, and construct a logarithmic barrier function B(x,ξ)=-ln((g(x)+ξ) / ξ), where g(x) represents the constraint function;

[0024] By adjusting the value of the auxiliary state variable ξ, the steepness of the barrier function is dynamically changed, thereby generating an adaptive barrier function.

[0025] The barrier function parameters are optimized using gradient descent, including:

[0026] Construct a composite Lyapunov function V(x,ξ) and design an adaptive law ξ'=-γ(B(x,ξ)-α), where γ is the adaptive gain with a value range of 1-10 and α is the desired barrier function value;

[0027] The parameters are updated using stochastic gradient descent, with an update step size of 0.01–0.1.

[0028] The second control variable is generated by introducing a momentum coefficient of 0.9 and an adaptive learning rate strategy.

[0029] Construct a control model for the T-product dynamic system, including:

[0030] The joint angle data, angular velocity data, motor torque data, and end-effector pose data of the painting robot within the sampling period T are collected as input data.

[0031] The system state space is divided into N subspaces, where N ranges from 4 to 8, and each subspace corresponds to a local linear model.

[0032] We select cubic B-spline basis functions as basis functions, construct the tensor representation of the state transition matrix, and generate the T-product dynamic system model structure.

[0033] Enables online identification and updating of model parameters, including:

[0034] Based on the T-product dynamic system model structure, the Tucker decomposition method is used to decompose the high-dimensional parameter identification problem into three-dimensional sub-problems.

[0035] A weighted least squares optimization problem is constructed, with weight coefficients set to 0.1-1.0, and the L2 regularization method is used to solve for the parameters of each local model.

[0036] The model parameters are updated using a recursive least squares algorithm, with the exponential forgetting factor set to 0.95-0.99.

[0037] The model structure is adjusted based on a sliding time window of 100-500 sampling points, with the parameter change rate limited to within 20%, to generate the third control variable.

[0038] A hierarchical control strategy is adopted to achieve collaborative control of multiple robots, including:

[0039] Based on the aforementioned third control variable, an improved Hungarian algorithm is used for task allocation, with optimization objectives including:

[0040] Minimize overall energy consumption, ensure that the workload difference between each robot is less than 15%, and control the coating overlap rate to 5%-10%;

[0041] An improved RRT* algorithm is used at the global layer to generate collision-free paths, with the sampling point spacing set to 50-100 mm;

[0042] The trajectory is optimized using a fifth-order polynomial function at the local layer, with the position error limited to ±0.1 mm and the attitude error limited to ±0.1 degrees.

[0043] To achieve collaborative control of multiple robots, including:

[0044] A hierarchical control architecture is adopted, with a control cycle of 100 milliseconds for the upper-layer collaborative controller and 1 millisecond for the lower-layer independent controller.

[0045] The coupling relationship between robots is established using the virtual structure method, with the coupling stiffness coefficient set to 10-100;

[0046] Design an event-triggered distributed control strategy, with the trigger threshold set to a position error of 0.5 mm or an attitude error of 0.5 degrees;

[0047] A prediction compensation mechanism of 100-200 milliseconds is introduced to ensure the stability of the system when the communication latency and data loss rate do not exceed 5%.

[0048] This application also provides an intelligent multi-task collaborative operation system for spray painting robots, including:

[0049] The data acquisition module is used to collect joint angles, angular velocities, torque data, and control parameters of the painting robot;

[0050] The discretized sliding mode control module is used to receive data from the data acquisition module and generate a first control quantity. The discretized sliding mode control module is connected to the adaptive barrier function module through a data bus.

[0051] An adaptive barrier function module is used to receive the first control quantity and generate a second control quantity in combination with robot system constraints. The adaptive barrier function module is connected to the dynamic model module through a real-time data interface.

[0052] The dynamic model module is used to receive the second control quantity, perform online parameter identification, and generate the third control quantity. The dynamic model module is connected to the cooperative control module via Ethernet.

[0053] The collaborative control module is used to receive the third control quantity, plan the collaborative trajectory of multiple robots, and output control commands. The collaborative control module is connected to each robot controller through a CAN bus to complete the multi-task collaborative operation of the painting robot.

[0054] This application embodiment also provides a computer device, the computer device comprising:

[0055] At least one processor; and,

[0056] A memory communicatively connected to the at least one processor; wherein,

[0057] The memory stores instructions that can be executed by the at least one processor, which enables the at least one processor to perform the above-described intelligent painting robot multi-task collaborative operation method.

[0058] This application also provides a computer-readable storage medium that stores computer instructions for causing a computer to execute the above-described intelligent spraying robot multi-task collaborative operation method.

[0059] This application also provides a computer program product, including computer instructions, which, when executed by a processor, implement the steps of the above-described intelligent spraying robot multi-task collaborative operation method.

[0060] This application has the following technical effects:

[0061] 1. By designing a discrete sliding mode controller that takes into account the influence of the sampling period, the problem of decreased accuracy in traditional discrete implementations is solved, and the control accuracy is improved;

[0062] 2. Introducing auxiliary state variables to adaptively adjust the barrier function parameters improves the system's adaptability to environmental changes and enhances its robustness;

[0063] 3. The T-product dynamic system modeling method is adopted to achieve high-precision modeling of complex nonlinear systems and provide accurate system model support;

[0064] 4. The multi-robot collaborative control system based on the above technology can efficiently complete complex spraying tasks, improving production efficiency and spraying quality. Attached Figure Description

[0065] To more clearly illustrate the technical solutions of the embodiments of this disclosure, the accompanying drawings used in the embodiments will be briefly described below. These drawings are incorporated in and constitute a part of this specification. They illustrate embodiments conforming to this disclosure and, together with the specification, serve to explain the technical solutions of this disclosure. It should be understood that the following drawings only show some embodiments of this disclosure and should not be considered as limiting the scope. Those skilled in the art can obtain other related drawings based on these drawings without creative effort.

[0066] Figure 1 A flowchart of the intelligent spraying robot multi-task collaborative operation method provided in the embodiments of this application;

[0067] Figure 2 This is an architecture diagram of a multi-robot collaborative control system provided in an embodiment of this application. Detailed Implementation

[0068] To make the objectives, technical solutions, and advantages of the embodiments of this disclosure clearer, the technical solutions of the embodiments of this disclosure will be clearly and completely described below with reference to the accompanying drawings. Obviously, the described embodiments are only a part of the embodiments of this disclosure, and not all of them. The components of the embodiments of this disclosure described and shown in the accompanying drawings can generally be arranged and designed in various different configurations. Therefore, the following detailed description of the embodiments of this disclosure provided in the accompanying drawings is not intended to limit the scope of the claimed disclosure, but merely represents selected embodiments of this disclosure. All other embodiments obtained by those skilled in the art based on the embodiments of this disclosure without inventive effort are within the scope of protection of this disclosure.

[0069] like Figure 1 As shown in the figure, this application provides an intelligent spraying robot multi-task collaborative operation method, including:

[0070] S1: Collect the joint angles, angular velocities, torque data and control parameters of the painting robot as state data, establish a discrete state equation containing the sampling period T based on the state data, introduce a sampling period compensation term to design a sliding mode switching surface, construct a discrete Lyapunov function, and generate the first control quantity.

[0071] S1 includes:

[0072] S1.1: Perform Taylor expansion on the robot dynamics equations at the sampling time, retain the second-order terms, and establish discrete state equations containing the sampling period T;

[0073] When constructing the discrete state equations, the continuous dynamics model of the painting robot must first be considered. This model typically includes the inertia matrix, Coriolis force matrix, and gravity term. Traditional methods often directly use the Euler method for discretization, neglecting the influence of higher-order terms. This method innovatively employs Taylor series expansion and retains second-order terms, enabling the discrete model to more accurately describe the system's dynamic characteristics. Specifically, at each sampling time, the derivatives of the state variables are Taylor-expanded, and then the second-order terms are truncated to obtain a discrete state equation containing the sampling period T. This approach effectively compensates for model errors introduced by sampling discretization, providing a more accurate system model for subsequent controller design.

[0074] S1.2: Based on the discrete state equation, design an improved sliding mode switching surface and construct a quadratic discrete Lyapunov function;

[0075] After obtaining the discrete state equations, an appropriate sliding mode switching surface needs to be designed to guide the system state to converge to the desired trajectory. Traditional sliding mode switching surface designs typically do not consider the impact of discrete sampling, leading to a decrease in practical control performance. To address this issue, this method designs an improved sliding mode switching surface, which explicitly introduces a sampling period compensation term. This compensation term can predict the system state change at the next sampling time and adjust the current control strategy accordingly, effectively reducing the tracking error caused by discrete implementation. After designing the sliding mode switching surface, a quadratic discrete Lyapunov function based on the switching surface is constructed to provide a theoretical tool for subsequent stability analysis. This form of Lyapunov function has good mathematical properties, facilitating the derivation of stability conditions.

[0076] S1.3: Analyze the forward difference of the discrete Lyapunov function to obtain the stability condition;

[0077] To ensure system stability, it is necessary to analyze the changes of the constructed discrete Lyapunov function at adjacent sampling times. By calculating the forward difference of the Lyapunov function, i.e., the difference between the function value at the current time and the next time, sufficient conditions for system stability can be derived. In the discrete domain, stability conditions typically require the forward difference of the function to be negative or to satisfy some decreasing relationship. Based on the obtained discrete state equations and the improved sliding mode switching surface, the forward difference expression is derived in detail, and the stability conditions that the controller design must satisfy are obtained through mathematical transformations. These conditions are directly related to the selection of controller parameters and are crucial for ensuring stable system operation.

[0078] S1.4: Based on the stability condition, design a discrete sliding mode controller that includes an equivalent control term and a switching control term, and generate the first control quantity;

[0079] Based on the stability conditions derived above, a complete discrete sliding mode controller structure is designed. This controller comprises two parts: an equivalent control term and a switching control term. The equivalent control term primarily compensates for the deterministic dynamics of the system, ensuring the system state remains stable on the sliding mode switching surface under undisturbed conditions. The switching control term primarily counteracts uncertainties and external disturbances in the system, enhancing its robustness. The combined effect of these two control variables ensures good tracking performance and anti-interference capability under various operating conditions. By rationally designing the controller structure and parameters to ensure the stability conditions derived above are met, the first control variable is generated as the initial control input to the system.

[0080] S1.4 includes:

[0081] S1.4.1: Based on the stability conditions, design an equivalent control term to compensate for the known dynamics of the system, and design a switching control term to counteract external disturbances;

[0082] The design of the equivalent control term is based on the deterministic dynamic model of the system. By solving the conditions that keep the system state on the sliding mode switching surface, the control input that compensates for the known dynamics of the system is obtained. Specifically, the discrete state equations of the system are substituted into the expression of the sliding mode switching surface, and the calculation formula of the equivalent control term is derived mathematically. This control term can effectively compensate for the nonlinear dynamics of the system, including the effects of factors such as inertia, Coriolis force, and gravity. The switching control term mainly targets the uncertainties and external disturbances in the system. Its design principle is to ensure that the system state can still be attracted to the vicinity of the sliding mode switching surface under the maximum disturbance amplitude. By reasonably setting the control gain, it is ensured that the switching control term has sufficient control strength to counteract various disturbance factors. S1.4.2: In the discrete sliding mode controller, a continuous reaching law is adopted, and the boundary layer thickness is set to a value in the range of 0.01-0.1.

[0083] The sign function commonly used in traditional sliding mode control leads to high-frequency jitter in the control input, adversely affecting the system's actuators. To address this issue, this method employs a continuous reaching law instead of the traditional discontinuous switching strategy. The continuous reaching law introduces the concept of a boundary layer, creating a smooth transition region near the sliding mode switching surface, resulting in smoother changes in the control input. Based on extensive experimental data and theoretical analysis, setting the boundary layer thickness within the range of 0.01-0.1 effectively reduces system jitter while maintaining control performance. The specific value of the boundary layer thickness can be fine-tuned according to the actual application scenario; smaller values ​​improve control accuracy, while larger values ​​help reduce jitter.

[0084] S1.4.3: Use a saturation function to replace the sign function in the switching control item to generate the first control quantity;

[0085] To further reduce control jitter, a saturation function is used instead of the traditional sign function in the switching control term. The saturation function exhibits a linear variation near the sliding mode switching surface and remains fixed beyond the boundary layer. This characteristic makes the control input change more smoothly as the system state approaches the target trajectory. Specifically, a saturation function is defined with the ratio of the switching surface value to the boundary layer thickness as the independent variable. When the absolute value of this ratio is less than 1, the function output is proportional to the input; when the absolute value is greater than or equal to 1, the function output is ±1. This design retains the robustness of sliding mode control while effectively solving the chattering problem. By reasonably combining the equivalent control term and the switching control term using the saturation function, a smooth and effective first control quantity is generated, providing the basic input for subsequent control stages.

[0086] S2: Based on the first control variable, combined with robot joint angle constraints, speed constraints, acceleration constraints and workspace boundary constraints, an adaptive barrier function is constructed, and the barrier function parameters are optimized using the gradient descent method to generate the second control variable;

[0087] S2 includes:

[0088] S2.1: Set the robot joint angle limit to ±170 degrees, the speed limit to ±180 degrees / second, and the acceleration limit to ±900 degrees / square second;

[0089] Before constructing the adaptive barrier function, it is necessary to clearly define the system's constraints. For a painting robot system, the most basic physical constraints include joint angle limits, speed limits, and acceleration limits. Based on the robot's mechanical structure and actuator characteristics, the joint angle limit is set to ±170 degrees. This setting considers both the physical range of motion of the robotic arm and provides a 10-degree safety margin to prevent instability that may occur when the robotic arm approaches its limit position. The speed limit is set to ±180 degrees / second. This value is determined based on the maximum speed of the motor and the reduction ratio to ensure that the system maintains precise control even at high speeds. The acceleration limit is set to ±900 degrees / square second. This value considers the dynamic response characteristics of the mechanical structure and the smoothness requirements of the painting process, avoiding excessive acceleration that could lead to mechanical vibration or uneven painting. These constraints together form the basis for the subsequent barrier function design.

[0090] S2.2: Introduce an auxiliary state variable ξ with a value range of 0.1-10, and construct a logarithmic barrier function B(x,ξ)=-ln((g(x)+ξ) / ξ), where g(x) represents the constraint function;

[0091] Traditional barrier functions typically have a fixed shape, making dynamic adjustment based on system state difficult. To address this issue, this method innovatively introduces an auxiliary state variable ξ, with a value range of 0.1-10, to dynamically adjust the characteristics of the barrier function. Based on this auxiliary variable, a logarithmic barrier function B(x,ξ) = -ln((g(x)+ξ) / ξ) is constructed, where g(x) represents the constraint function, reflecting the distance between the system state and the constraint boundary. This barrier function structure possesses special properties: when the system state is close to the constraint boundary, g(x) approaches zero, and the barrier function value increases rapidly, generating a strong repulsive effect; when the system state is far from the constraint boundary, the barrier function value is small, with limited impact on the system. By introducing the auxiliary variable ξ, the steepness and range of the barrier function can be dynamically adjusted, ensuring that the system satisfies the constraint requirements while maintaining good performance.

[0092] S2.3: By adjusting the value of the auxiliary state variable ξ, the steepness of the barrier function is dynamically changed to generate an adaptive barrier function;

[0093] Adjusting the auxiliary state variable ξ is key to achieving the adaptive characteristics of the barrier function. When the system state is far from the constraint boundary, the value of ξ can be set larger. In this case, the gradient of the barrier function is smaller, allowing the system to move more freely, which is beneficial to improving control performance. When the system state is close to the constraint boundary, the value of ξ is decreased, making the barrier function steeper near the constraint boundary, generating a stronger repulsive force, and ensuring that the constraint conditions are not violated. This dynamic adjustment strategy allows the barrier function to automatically adjust its characteristics according to changes in the system state, ensuring that the constraints are satisfied while minimizing the impact on system performance. By rationally designing the adjustment mechanism of ξ, the barrier function acquires adaptive capabilities, enabling it to adapt to different operating conditions and task requirements, forming a truly adaptive barrier function.

[0094] S2.4: Construct a composite Lyapunov function V(x,ξ) and design an adaptive law ξ'=-γ(B(x,ξ)-α), where γ is the adaptive gain, with a value range of 1-10, and α is the desired barrier function value;

[0095] To design an adaptive adjustment rule for the auxiliary state variable ξ, a composite Lyapunov function V(x,ξ) containing the system state and the auxiliary variable needs to be constructed. This function comprehensively considers the stability of the system state and the constraint satisfaction, providing a theoretical basis for adaptive adjustment. Based on this composite Lyapunov function, the adaptive law for the auxiliary variable is designed as ξ'=-γ(B(x,ξ)-α), where γ is the adaptive gain, with a value range of 1-10, and α is the desired barrier function value. This design follows the gradient descent principle: when the current barrier function value is higher than the desired value, the value of ξ increases, weakening the influence of the barrier function; when the barrier function value is lower than the desired value, the value of ξ decreases, enhancing the effect of the barrier function. The adaptive gain γ controls the adjustment speed; a larger γ value makes the system response faster but may lead to over-adjustment; a smaller γ value makes the adjustment process smoother but the response slower. In practical applications, the value of γ can be adjusted according to system characteristics and task requirements to obtain optimal performance.

[0096] S2.5: Update the parameters using stochastic gradient descent, with a parameter update step size of 0.01-0.1;

[0097] To optimize the performance of the barrier function, in addition to adaptive adjustment of auxiliary variables, the parameters of the barrier function itself also need to be optimized. This method uses stochastic gradient descent for parameter updates, which is computationally efficient, easy to implement, and suitable for online applications. During the optimization process, an appropriate performance index function is first defined, which typically includes measures of both control precision and constraint satisfaction. Then, the gradient of the performance index function with respect to each parameter is calculated, and the parameters are updated according to the gradient direction. To ensure the stability of the optimization process, the parameter update step size is set in the range of 0.01-0.1. Smaller step sizes make the optimization process smoother but result in slower convergence; larger step sizes can accelerate convergence but may lead to parameter oscillations. In practical applications, the step size can be dynamically adjusted according to the system's response characteristics and the urgency of the task to balance the requirements of convergence speed and stability.

[0098] S2.6: Introduce a momentum coefficient of 0.9 and an adaptive learning rate strategy to generate the second control variable;

[0099] To further improve the efficiency and stability of parameter optimization, a momentum term and an adaptive learning rate strategy are introduced based on the stochastic gradient descent method. The momentum coefficient is set to 0.9, which ensures that parameter updates consider not only the current gradient but also historical update directions, helping to overcome local optima and accelerate the convergence process. The adaptive learning rate strategy dynamically adjusts the learning rate based on gradient changes during optimization. When gradient directions are consistent multiple times, the learning rate is appropriately increased to accelerate convergence; when gradient directions change frequently, the learning rate is decreased to improve stability. Through these improved strategies, the parameter optimization process becomes more efficient and robust, quickly adapting to changes in system state and environment. Based on the optimized barrier function parameters and the adaptively adjusted auxiliary variables, combined with the first control variable from the previous step, a second control variable that comprehensively considers the constraints is generated, providing input for subsequent control loops.

[0100] S3: Based on the second control quantity, collect the input and output data of the spraying robot during online operation, construct a T-product dynamic system control model, realize online identification and updating of model parameters, and generate a third control quantity;

[0101] S3 includes:

[0102] S3.1: Collect the joint angle data, angular velocity data, motor torque data and end-effector pose data of the painting robot within the sampling period T as input data;

[0103] Before constructing the data-driven T-product dynamic system control model, comprehensive data acquisition is essential. This method employs a multi-dimensional, high-precision data acquisition strategy to systematically collect various key data during the operation of the painting robot. Specifically, the robot's control system uses built-in encoders to collect angle data for each joint, directly reflecting the robot's posture information; angular velocity sensors acquire angular velocity data for each joint, reflecting the robot's dynamic characteristics; the motor driver's current detection circuit indirectly obtains motor torque data, reflecting the robot's dynamic characteristics during motion; and forward kinematics calculations or an external vision system acquire pose data for the robot's end effector, reflecting the robot's actual position and posture in the workspace. During data acquisition, particular attention is paid to the consistency of the sampling period to ensure that all data accurately corresponds to the same point in time, providing high-quality foundational data for subsequent model construction.

[0104] S3.2: Divide the system state space into N subspaces, where N ranges from 4 to 8, and each subspace corresponds to a local linear model;

[0105] Traditional global nonlinear models often fail to accurately describe the dynamic characteristics of complex systems, while simple linear models lack expressive power. To resolve this contradiction, this method employs piecewise linearization, dividing the entire state space of the system into multiple subspaces. Based on the complexity and nonlinearity of the painting robot system, the state space is divided into 4 to 8 subspaces, a range determined through extensive experimental and theoretical analysis. Too few subspaces result in each subspace having an excessively large coverage area, leading to insufficient accuracy of the local linear models; too many subspaces result in overly complex models, increasing computational burden and potentially causing overfitting. In practical partitioning, uniform partitioning or non-uniform partitioning based on data density can be used to ensure that the system behavior within each subspace can be well approximated by a linear model. Each subspace corresponds to a local linear model, and these local models collectively form the complete system model.

[0106] S3.3: Select cubic B-spline basis functions as basis functions, construct the tensor representation of the state transition matrix, and generate the T-product dynamic system model structure;

[0107] To achieve smooth transitions between subspaces and continuity of the overall model, this method selects cubic B-spline basis functions as the basic building blocks. Cubic B-spline basis functions possess good smoothness and local support, enabling them to reduce abrupt changes at subspace boundaries while ensuring model continuity. Based on these basis functions, a tensor representation of the state transition matrix is ​​constructed. Specifically, the system's state transition relationship is represented as a weighted combination of multiple basis functions, with these weight parameters forming a high-dimensional tensor. Each dimension of the tensor corresponds to an aspect of the system state, such as joint angles or angular velocities. This representation method maintains the structural simplicity of the model while possessing the ability to express complex nonlinear relationships. Through the rational design of the tensor structure and dimensional partitioning, a complete T-product dynamic system model framework is formed. This framework can uniformly describe the dynamic behavior of the system under different operating conditions, providing a structural foundation for subsequent parameter identification.

[0108] S3.4: Based on the T-product dynamic system model structure, the Tucker decomposition method is used to decompose the high-dimensional parameter identification problem into three-dimensional sub-problems;

[0109] The parameters in the T-product dynamic system model exhibit a high-dimensional tensor structure. Directly identifying the parameters of the entire tensor is computationally intensive and inefficient. To address this issue, this method employs Tucker decomposition to break down the high-dimensional parameter identification problem into multiple low-dimensional subproblems. Tucker decomposition is a generalized tensor decomposition method that can decompose a high-dimensional tensor into a product of a core tensor and several factor matrices. In this application, the parameter tensor is decomposed into subproblems in three main dimensions, corresponding to the robot's configuration space, velocity space, and control input space, respectively. Through this decomposition, the large number of parameters that originally needed to be identified simultaneously are divided into several relatively independent parameter sets, significantly reducing computational complexity. During the decomposition process, an approximate model is first constructed based on the initial data. Then, the factor matrices are iteratively optimized using alternating least squares methods to obtain the complete decomposition result. This method not only improves computational efficiency but also enhances the stability and noise resistance of parameter identification.

[0110] S3.5: Construct a weighted least squares optimization problem with weight coefficients set to 0.1-1.0, and use the L2 regularization method to solve for the parameters of each local model;

[0111] After obtaining the sub-problems of each dimension of the Tucker decomposition, it is necessary to accurately identify the parameters of each local linear model. This method constructs a weighted least squares optimization problem, determining the optimal parameter values ​​by minimizing the prediction error. To handle the differences in importance among different data points, a weighting coefficient mechanism is introduced, with the weights ranging from 0.1 to 1.0. Data points closer to the current working point are assigned higher weights to ensure higher accuracy of the local model within the working region; while data points farther away are assigned lower weights to reduce their influence. Simultaneously, to prevent overfitting and improve the model's generalization ability, L2 regularization is employed, adding a parameter norm penalty term to the optimization objective function. The introduction of L2 regularization makes the model parameters tend to take smaller values, enhancing the smoothness and stability of the model. The optimization problem is solved numerically, such as the conjugate gradient method or the quasi-Newton method, ultimately obtaining the optimal parameter estimates for each local model. In this way, the fitting accuracy of the local model is guaranteed while avoiding the risk of overfitting.

[0112] S3.6: The model parameters are updated using a recursive least squares algorithm, with the exponential forgetting factor set to 0.95-0.99;

[0113] The dynamic characteristics of a painting robot system may change over time; factors such as mechanical wear and changes in ambient temperature can affect system behavior. To enable the model to adapt to these changes, this method employs a recursive least squares algorithm to update model parameters online. The core idea of ​​this algorithm is that when new data arrives, it does not need to reprocess all historical data, but instead directly calculates updated parameter estimates based on previous parameter estimates and the information increment of the new data. To balance the influence of historical and new data, an exponential forgetting factor is introduced, with a value range of 0.95 to 0.99. A smaller forgetting factor results in a faster response to new data, but may lead to increased volatility in parameter estimates; a larger forgetting factor makes the algorithm more stable, but slower to respond to system changes. In practical applications, the value of the forgetting factor can be dynamically adjusted according to system characteristics and the rate of environmental change. Through the online update mechanism of the recursive least squares algorithm, the model can continuously track the dynamic changes of the system and maintain good prediction accuracy.

[0114] S3.7: Adjust the model structure based on a sliding time window of 100-500 sampling points, limit the parameter change rate to within 20%, and generate the third control quantity;

[0115] In addition to online parameter updates, the model structure itself also needs to be adaptively adjusted according to changes in system behavior. This method designs a model structure adjustment mechanism based on a sliding time window, with the window size set between 100 and 500 sampling points. A smaller window size allows the model to respond quickly to changes in system behavior but may be more sensitive to noise and interference; a larger window size provides more stable estimates but responds more slowly to system changes. Based on the data within the sliding window, the applicability of the current model structure is periodically evaluated, including subspace partitioning and basis function selection. When the evaluation index falls below a preset threshold, the model structure is re-optimized. To ensure the stability of system control, the rate of parameter change in each adjustment is limited to no more than 20%, avoiding drastic fluctuations in control output due to model mutations. Finally, based on the updated model parameters and optimized model structure, combined with the second control variable from the previous step, a more accurate third control variable is generated, providing a reliable foundation for multi-robot collaborative control. This control model with adaptive learning capabilities can effectively cope with various operating condition changes and uncertainties, maintaining high-performance system operation.

[0116] S4: Based on the third control variable, establish a multi-robot kinematics and dynamics model, adopt a hierarchical control strategy to realize multi-robot cooperative control, output control commands that meet the requirements of workload balance and spraying quality, and control multiple spraying robots to complete cooperative operations.

[0117] S4 includes:

[0118] S4.1: An improved Hungarian algorithm is used for task allocation. The optimization objectives include: minimizing overall energy consumption, ensuring that the workload difference between each robot is less than 15%, and controlling the coating overlap rate to 5%-10%.

[0119] The efficiency of a multi-robot collaborative painting system largely depends on the rationality of task allocation. This method employs an improved Hungarian algorithm for task allocation, a classic combinatorial optimization method capable of finding the optimal solution to the assignment problem. Building upon the traditional Hungarian algorithm, a multi-objective optimization mechanism is introduced, simultaneously considering three key factors: energy minimization, load balancing, and painting quality. Energy minimization is primarily achieved by optimizing the robot's motion trajectory and work area, minimizing unnecessary large movements and acceleration / deceleration processes. Load balancing ensures that the workload difference between robots is controlled within 15%, avoiding situations where some robots are overloaded while others are idle, thus improving overall system efficiency. The painting overlap rate is controlled within the range of 5%-10%, ensuring that there are no blank areas on the workpiece surface while avoiding material waste and uneven surface quality caused by excessive overlap. In the algorithm implementation, the entire painting task is first divided into multiple sub-tasks, then a cost matrix for task allocation is constructed, and finally, the optimal allocation scheme is solved through iterative optimization. This multi-objective optimization task allocation strategy lays a solid foundation for subsequent collaborative control.

[0120] S4.2: An improved RRT* algorithm is used at the global layer to generate collision-free paths, with the sampling point spacing set to 50-100 mm;

[0121] After task allocation, safe and efficient motion paths need to be planned for each robot. This method employs a hierarchical planning strategy. First, at the global level, an improved RRT* (Rapidly-exploring Random Tree Star) algorithm is used to generate a collision-free basic path. The RRT* algorithm is a random sampling path planning method with the ability to efficiently explore complex environments. Based on the traditional RRT* algorithm, this method introduces a heuristic sampling strategy and a dynamic search radius adjustment mechanism, improving the algorithm's performance in the spraying environment. The heuristic sampling strategy adjusts the distribution of sampling points according to the workpiece surface features and spraying requirements, making the algorithm more inclined to conduct detailed exploration in critical areas. The dynamic search radius is automatically adjusted according to the environmental complexity, using a larger radius in open areas to accelerate convergence and a smaller radius in narrow areas to improve accuracy. The sampling point spacing is set to 50-100 mm, a range determined by a combination of spraying process requirements and computational efficiency. Too small a spacing leads to a dramatic increase in computation with limited accuracy improvement, while too large a spacing may miss important path selections. Through this improved RRT* algorithm, a global path that meets the collision-free requirement can be efficiently generated, providing a foundation for subsequent trajectory optimization.

[0122] S4.3: The trajectory is optimized using a fifth-order polynomial function in the local layer, with the position error limited to ±0.1 mm and the attitude error limited to ±0.1 degrees;

[0123] Global path planning primarily focuses on generating collision-free paths, while local trajectory optimization emphasizes improving path smoothness and dynamic feasibility to meet the high requirements of coating quality. This method employs a fifth-order polynomial function for trajectory optimization at the local level. The fifth-order polynomial is chosen because it ensures the continuity of velocity and acceleration while providing sufficient degrees of freedom to satisfy boundary conditions. During optimization, key points of the global path are used as constraints, and an optimization objective function is constructed by combining the robot's dynamic characteristics and coating process requirements. This function typically includes multiple performance indicators, such as trajectory length, energy consumption, and coating uniformity. Optimal parameters are solved using numerical optimization methods to generate a smooth trajectory that satisfies all constraints. Regarding accuracy control, position error is limited to ±0.1 mm and attitude error to ±0.1 degrees. This accuracy requirement is determined based on the demands of a high-quality coating process. To achieve this accuracy target, the trajectory optimization process specifically considers the robot's kinematic errors, dynamic nonlinearities, and the response characteristics of the coating system. A combination of feedforward compensation and feedback correction ensures high-precision trajectory tracking.

[0124] S4.4: Adopts a hierarchical control architecture with a control cycle of 100 milliseconds for the upper-level collaborative controller and 1 millisecond for the lower-level independent controller;

[0125] The complexity of multi-robot cooperative control systems necessitates a hierarchical control architecture, dividing system functions according to time scale and abstraction level. This approach designs a two-layer control architecture: the upper layer is a cooperative controller, primarily responsible for coordination and synchronization among multiple robots, ensuring coordinated execution of the overall task; the lower layer consists of independent controllers for each robot, responsible for precise trajectory tracking and motion execution. The control cycle settings of the two layers differ significantly: the upper-layer cooperative controller has a control cycle of 100 milliseconds, a setting that considers the computational complexity and communication latency of multi-robot coordination decisions while meeting the time response requirements of the cooperative task; the lower-layer independent controllers have a control cycle of 1 millisecond, this high-frequency control ensures precise execution of individual robot actions and timely response to dynamic changes and external disturbances. The two layers of controllers interact through a bidirectional communication mechanism: the upper-layer controller sends task instructions and coordination signals to the lower layer, while the lower-layer controller feeds back execution status and environmental information to the upper layer. This hierarchical control architecture ensures both overall system coordination and maintains high precision and real-time performance of individual robot control.

[0126] S4.5: Establish the coupling relationship between robots using the virtual structure method, and set the coupling stiffness coefficient to 10-100;

[0127] To achieve coordinated movement of multiple robots, it is necessary to establish coupling relationships between them, enabling them to work collaboratively as a whole. This method employs a virtual structure approach to construct such coupling relationships. The core idea of ​​this method is to treat multiple robots as nodes connected to a virtual rigid structure, and to achieve coordinated control of the individual robots by controlling the overall movement of the virtual structure. In the specific implementation process, the topological relationship of the virtual structure is first defined, and the relative positions of each robot in the virtual structure are determined according to task requirements and workpiece characteristics. Then, a virtual spring-damping system is designed to connect each robot to the virtual structure, forming an elastically coupled network. The coupling stiffness coefficient is a key parameter for controlling the coupling strength, and is set within the range of 10-100. A lower stiffness coefficient makes the system more flexible, giving each robot more independent adaptation space, suitable for handling workpieces with complex shapes or requiring local adjustments; a higher stiffness coefficient makes the system more rigid, resulting in more synchronized movement of the robots, suitable for precision painting tasks requiring strict coordination. In practical applications, the coupling stiffness can be dynamically adjusted according to the characteristics of different workpieces and painting requirements to achieve the best collaborative effect.

[0128] S4.6: Design an event-triggered distributed control strategy, with the trigger threshold set to a position error of 0.5 mm or an attitude error of 0.5 degrees;

[0129] Traditional time-triggered control strategies require communication and computation within fixed time intervals, easily leading to wasted network resources and excessive computational burden. To improve system efficiency, this method designs an event-triggered distributed control strategy, where control calculations and information interaction only occur when the system state changes significantly. The trigger condition is set at a position error of 0.5 mm or an attitude error of 0.5 degrees; this threshold is determined comprehensively based on the coating quality requirements and the system's dynamic characteristics. Setting the threshold too low will lead to frequent triggering, negating the resource-saving advantage of event triggering; setting it too high may affect control performance. During actual operation, each robot independently monitors its own state. When the error exceeds the trigger threshold, it sends a state update request to the collaborative controller and receives the updated control instructions. Simultaneously, a maximum no-trigger time limit is introduced to ensure that even if the system remains in a stable state for a long time, information can still be updated periodically, preventing collaborative failure due to communication interruptions. This event-triggered distributed control strategy significantly reduces the system's communication burden and computational resource consumption while maintaining good control performance.

[0130] S4.7: Introduces a 100-200 millisecond prediction compensation mechanism to ensure system stability when communication latency and data loss rate do not exceed 5%.

[0131] In practical multi-robot collaborative systems, communication latency and data loss are unavoidable problems, which can severely impact system stability and control performance. To address these challenges, this method introduces a predictive compensation mechanism. By predicting the system state 100-200 milliseconds in the future, control commands are generated in advance, effectively compensating for the impact of communication latency. The prediction time range is set considering the typical communication latency and controller response time of the system. Too short a prediction time makes it difficult to effectively compensate for latency, while too long a prediction time may lead to inaccurate predictions due to accumulated errors. The prediction model is based on the previously constructed T-product dynamic system model, combined with a Kalman filter to achieve state estimation and prediction. Simultaneously, a packet loss handling mechanism is designed. When data loss is detected, the system generates an estimate based on historical data and the prediction model, maintaining control continuity. Through theoretical analysis and experimental verification, this predictive compensation mechanism ensures that the system maintains stable operation even with communication latency and data loss rates not exceeding 5%. This characteristic is particularly important for multi-robot collaborative systems in real-world industrial environments, significantly improving system robustness and reliability, and providing a guarantee for the completion of high-quality painting tasks.

[0132] In another embodiment, the steps of the intelligent spraying robot multi-task collaborative operation method include:

[0133] First, in constructing the discretized sliding mode control framework, it is necessary to establish the discrete state equations of the system. Considering the influence of the sampling period T in the actual control system, the continuous system model is discretized. Specifically, the robot's dynamic equations are Taylor-expanded at sampling time tk, and higher-order terms are retained to obtain more accurate discrete state equations. This approach differs from the traditional direct truncation method and can better preserve the dynamic characteristics of the system.

[0134] For a painting robot system, its dynamic equations can be expressed as:

[0135]

[0136] Where M(q) is the inertia matrix, Let G(q) be the Coriolis force and centrifugal force matrix, G(q) be the gravity matrix, and τ be the control input.

[0137] Define the system state as The state space of the system is then represented as:

[0138]

[0139] Where f(x) and g(x) are nonlinear functions, and u is the control input.

[0140] By Taylor expansion and retaining the second-order terms, we obtain the discrete state equations:

[0141] x(k+1)=x(k)+Tf(x(k),u(k))+(T 2 / 2)f'(x(k),u(k))f(x(k),u(k))+O(T 3 )

[0142] Where T is the sampling period, and f' represents the derivative of function f with respect to x.

[0143] After obtaining the discrete state equations, an improved sliding mode switching surface is designed. Traditional switching surface designs often neglect the influence of discrete sampling, which may lead to chattering in actual system operation. To solve this problem, a sampling period compensation term σ(k) is introduced into the switching surface design. The core idea of ​​this compensation term is to predict the trend of system state changes within each sampling period and adjust the switching surface parameters accordingly. The sliding mode switching surface is defined as:

[0144] σ(k)=s(k)+αTs(k+1|k)

[0145] Where s(k) is the conventional sliding mode switching surface, s(k+1|k) is the predicted value of the switching surface at the next time step, and α is the weighting coefficient.

[0146] To ensure that the designed controller can converge the system state to the sliding surface and maintain stable motion, an appropriate discrete Lyapunov function is constructed. A quadratic function based on the switching surface, V(k) = 1 / 2σ(k)², is chosen as a candidate Lyapunov function. By analyzing the forward difference ΔV(k) = V(k+1) - V(k) in the discrete domain, the controller design conditions that guarantee system stability are derived:

[0147] ΔV(k)=V(k+1)-V(k)≤-ηV(k)

[0148] Where η is a positive constant, representing the convergence rate.

[0149] Based on the above analysis, a complete discrete sliding mode controller is designed. The controller design includes both an equivalent control term ueq(k) to compensate for the known dynamics of the system and a switching control term us(k) to counteract system uncertainties and external disturbances. To reduce chattering in the discrete implementation, a continuous reaching law is used instead of the traditional sign function, and a boundary layer design is introduced. Specifically, the control law is expressed as:

[0150] u(k) = ueq(k) + us(k)

[0151] Where, ueq(k)=-(g(x(k))T g(x(k))) -1g(x(k))T[f(x(k))+(σ(k)-s(k)) / T]

[0152] us(k)=-K·sat(σ(k) / Φ)

[0153] Where K is the control gain matrix, sat(·) is the saturation function, and Φ is the boundary layer thickness, set in the range of 0.01-0.1.

[0154] The above design realizes a discrete sliding mode controller that takes into account the influence of the sampling period, effectively solving the problem of accuracy degradation in traditional discrete implementations.

[0155] Secondly, before designing the adaptive barrier function for the auxiliary variables, the system's constraints and performance indicators must be clearly defined. For the painting robot system, the main constraints include physical constraints such as robot joint angle limits (±170 degrees), speed limits (±180 degrees / second), acceleration limits (±900 degrees / square second), and workspace boundaries. These constraints can be mathematically represented as the inequality constraint g(x) ≤ 0.

[0156] To achieve constraint softening and adaptive adjustment, an auxiliary state variable ξ is innovatively introduced to construct an adaptive barrier function structure. Specifically, for each constraint, a logarithmic barrier function of the form B(x,ξ)=-ln((g(x)+ξ) / ξ) is constructed, where ξ is an auxiliary variable ranging from 0.1 to 10. The special feature of this structure is that by adjusting the value of ξ, the steepness of the barrier function can be dynamically changed, thereby achieving adaptive control of the constraint boundary.

[0157] Next, designing the adaptive law for the parameters is crucial for achieving dynamic adjustment of the barrier function. Using a method based on Lyapunov stability theory, an adaptive update law for the auxiliary variable ξ is designed. Specifically, a composite Lyapunov function V(x,ξ) containing the system state and the auxiliary variable is constructed, and the adaptive law for ξ is derived by solving the condition dV / dt≤0:

[0158] ξ'=-γ(B(x,ξ)-α)

[0159] Where γ is the adaptive gain, ranging from 1 to 10, and α is the desired barrier function value.

[0160] To optimize the barrier function parameters and improve system performance, a gradient descent-based parameter optimization algorithm is established. By defining a suitable performance index function J(θ) (where θ represents the set of parameters to be optimized), the parameters are updated using stochastic gradient descent.

[0161]

[0162] Where η is the learning rate, ranging from 0.01 to 0.1. To improve optimization efficiency, a momentum term and an adaptive learning rate strategy are introduced:

[0163]

[0164] Through the above design, adaptive barrier function control based on auxiliary variables was realized, ensuring the constraint satisfaction. At the same time, the control strategy can be dynamically adjusted according to the system state, which significantly improves the system's adaptability and performance.

[0165] Secondly, when establishing the data-driven T-product dynamic system control model, the first step is to collect the system's input and output data. A multi-dimensional data acquisition strategy is adopted to simultaneously record the robot's joint angles, angular velocities, motor torques, end-effector poses, and other state variables, as well as control input signals and environmental parameters. Special attention is paid to the selection of the sampling frequency during data acquisition to ensure that the dynamic characteristics of the system are fully captured. To improve data quality, a filtering algorithm is used to preprocess the raw data, removing noise and outliers.

[0166] After obtaining a high-quality dataset, a T-product dynamic model structure is constructed. The core idea of ​​the T-product model is to decompose a nonlinear dynamic system into a tensor product of multiple locally linear models. Specifically, the system state space is first divided into N subspaces (N ranges from 4 to 8), with each subspace corresponding to a locally linear model. Then, by defining appropriate basis functions (cubic B-spline basis functions are chosen), a tensor representation of the state transition matrix is ​​constructed. The advantage of this structural design is that it maintains the nonlinear expressiveness of the model while facilitating subsequent parameter identification and controller design.

[0167] The design of the model parameter identification algorithm is a crucial step in ensuring the accuracy of the T-product model. A parameter identification method based on tensor decomposition is adopted, transforming the high-dimensional parameter identification problem into several low-dimensional sub-problems. Specifically, the high-dimensional parameter identification problem is first decomposed into three-dimensional sub-problems based on the Tucker decomposition method, obtaining the basic factors for each dimension. Then, a weighted least squares optimization problem is constructed, with weight coefficients set to 0.1-1.0, combined with L2 regularization, to solve for the parameters of each local model. To improve identification efficiency and accuracy, an adaptive step size strategy and an early stopping criterion are introduced to effectively avoid overfitting.

[0168] Finally, to enable the model to adapt to changes in system parameters and the working environment, an online model update mechanism was implemented. This mechanism consists of two main parts: first, fast parameter updates based on the Recursive Least Squares (RLS) algorithm; and second, adaptive adjustment of the model structure based on a sliding time window. During the online update process, an exponential forgetting factor is used to balance the influence of historical and new data, with the exponential forgetting factor set between 0.95 and 0.99. Simultaneously, the parameter change rate is limited to within 20% to ensure the smoothness of the model update. Furthermore, a sliding time window based on 100-500 sampling points is established for adaptive adjustment of the model structure. When the model accuracy drops to a preset threshold, a re-optimization of the model structure is triggered.

[0169] Through the above design, an adaptive T-product dynamic system model was realized. This design not only accurately describes the nonlinear dynamic characteristics of the system, but also maintains the timeliness of the model through an online update mechanism, providing reliable system model support for subsequent controller design.

[0170] Finally, in implementing the multi-robot cooperative control algorithm, it is first necessary to establish accurate kinematic and dynamic models of the multi-robots. For each painting robot, a modified DH parameter method is used to establish a kinematic model, including forward and inverse kinematic equations. Simultaneously, the Lagrangian method is used to derive the robot's dynamic equations, considering the effects of joint inertia, Coriolis force, gravity, and other factors. Specifically, special factors such as nozzle reaction force and the additional mass generated by paint flow during the painting process are also considered in the dynamic modeling. To improve model accuracy, a friction compensation model based on experimental data is introduced, including the effects of Coulomb friction, viscous friction, and static friction.

[0171] After completing the basic modeling, efficient task allocation and trajectory planning strategies were designed. Task allocation employed an improved Hungarian algorithm, considering multiple optimization objectives, including workload balancing, energy minimization, and optimal spraying quality. Specific objectives included minimizing overall energy consumption, ensuring workload differences among robots were less than 15%, and controlling spraying overlap within 5%-10%. Trajectory planning consisted of two levels: global planning and local planning. At the global level, an improved RRT* algorithm was used to generate collision-free coarse paths, with sampling point spacing set to 50-100 mm. At the local level, a trajectory optimization method based on fifth-order polynomials was employed to ensure trajectory smoothness and dynamic feasibility, limiting positional error to ±0.1 mm and attitude error to ±0.1 degrees. To improve planning efficiency, a parallel planning strategy based on workspace partitioning was designed, while a dynamic priority mechanism was introduced to handle potential conflicts between multiple robots.

[0172] Based on the previously developed discrete sliding mode controller, adaptive barrier function, and T-product dynamic model, a multi-robot cooperative control system was realized. The control system adopts a hierarchical architecture: the upper layer is the cooperative controller, responsible for the synchronization and coordination of multiple robots, with a control cycle of 100 milliseconds; the lower layer consists of independent controllers for each robot, achieving precise trajectory tracking, with a control cycle of 1 millisecond. The cooperative controller establishes the coupling relationship between robots through a virtual structure method, with the coupling stiffness coefficient set to 10-100, and uses a synchronization control algorithm based on consistency theory to ensure coordinated movement of multiple robots. To address practical problems such as communication delay and data loss, an event-triggered distributed control strategy was designed, with the trigger threshold set at a position error of 0.5 mm or an attitude error of 0.5 degrees. A prediction compensation mechanism of 100-200 milliseconds was introduced to ensure the system's stability when the communication delay and data loss rate do not exceed 5%.

[0173] Finally, to ensure the safe operation of the multi-robot system, a complete collision avoidance and safety protection mechanism was constructed. Collision avoidance employs a multi-layered approach: static obstacles are handled using an improved artificial potential field method, while dynamic obstacles (including other robots) are avoided in real-time using a velocity-based obstacle avoidance method. The safety protection mechanism comprises multiple layers: at the hardware level, emergency stop and limit protection are implemented; at the software level, real-time monitoring and fault diagnosis are achieved; and at the system level, a safety state machine is established, defining multiple safety levels and corresponding handling strategies. In particular, a deep learning-based anomaly detection algorithm was developed, capable of providing early warnings of potential safety hazards.

[0174] Through the above implementation methods, this application realizes an intelligent multi-task collaborative operation method and system for spraying robots. This method solves the problems of decreased discrete implementation accuracy, difficulty in adaptive adjustment of controller parameters, and lack of effective data-driven control strategies in the prior art by organically combining a discrete sliding mode controller design that considers the influence of sampling period, an adaptive barrier function for auxiliary variables, a data-driven T-product dynamic system control model, and a multi-robot collaborative control strategy. This enables efficient collaborative operation of multiple spraying robots.

[0175] Furthermore, this application also provides an intelligent multi-task collaborative operation system for a painting robot, including a data acquisition module, a discretized sliding mode control module, an adaptive barrier function module, a dynamic model module, and a collaborative control module. The modules interact with each other and transmit control signals through connections such as data bus, real-time data interface, Ethernet, and CAN bus, jointly completing the multi-task collaborative operation of the painting robot.

[0176] Specifically, such as Figure 2 As shown in the figure, this application provides an intelligent multi-task collaborative operation system for a painting robot, including:

[0177] The data acquisition module is used to collect joint angles, angular velocities, torque data, and control parameters of the painting robot;

[0178] The discretized sliding mode control module is used to receive data from the data acquisition module and generate a first control quantity. The discretized sliding mode control module is connected to the adaptive barrier function module through a data bus.

[0179] An adaptive barrier function module is used to receive the first control quantity and generate a second control quantity in combination with robot system constraints. The adaptive barrier function module is connected to the dynamic model module through a real-time data interface.

[0180] The dynamic model module is used to receive the second control quantity, perform online parameter identification, and generate the third control quantity. The dynamic model module is connected to the cooperative control module via Ethernet.

[0181] The collaborative control module is used to receive the third control quantity, plan the collaborative trajectory of multiple robots, and output control commands. The collaborative control module is connected to each robot controller through a CAN bus to complete the multi-task collaborative operation of the painting robot.

[0182] It should be noted that the parameter settings in each step of the above embodiments (such as boundary layer thickness, auxiliary state variable value range, adaptive gain, learning rate, exponential forgetting factor, etc.) are optimized values ​​derived from a large amount of experimental data and theoretical analysis. In practical applications, they can be appropriately adjusted according to specific working conditions to obtain the best control effect.

[0183] Furthermore, those skilled in the art will understand that the methods provided in the embodiments of this application are not only applicable to spraying robot systems, but can also be extended to other types of multi-robot collaborative operation scenarios, such as welding, assembly, and handling in industrial fields, and have broad application prospects.

[0184] Finally, it should be noted that the above-described embodiments are merely specific implementations of this disclosure, used to illustrate the technical solutions of this disclosure, and not to limit it. The protection scope of this disclosure is not limited thereto. Although this disclosure has been described in detail with reference to the foregoing embodiments, those skilled in the art should understand that any person skilled in the art can still modify or easily conceive of changes to the technical solutions described in the foregoing embodiments, or make equivalent substitutions for some of the technical features, within the scope of the technology disclosed in this disclosure; and these modifications, changes, or substitutions do not cause the essence of the corresponding technical solutions to deviate from the spirit and scope of the technical solutions of the embodiments of this disclosure, and should all be covered within the protection scope of this disclosure. Therefore, the protection scope of this disclosure should be determined by the protection scope of the claims.

Claims

1. A method for multi-task collaborative operation of an intelligent spraying robot, characterized in that, include: The joint angles, angular velocities, torque data, and control parameters of the painting robot are collected as state data. Based on the state data, a discrete state equation containing the sampling period T is established. A sampling period compensation term is introduced to design a sliding mode switching surface, a discrete Lyapunov function is constructed, and the first control quantity is generated. Based on the first control variable, combined with robot joint angle constraints, speed constraints, acceleration constraints, and workspace boundary constraints, an adaptive barrier function is constructed, and the barrier function parameters are optimized using the gradient descent method to generate the second control variable. Based on the second control variable, the input and output data of the spraying robot during online operation are collected to construct a T-product dynamic system control model, realize online identification and updating of model parameters, and generate a third control variable; Based on the third control variable, a multi-robot kinematics and dynamics model is established, and a hierarchical control strategy is adopted to realize multi-robot collaborative control. The output control commands meet the requirements of workload balance and spraying quality, and control multiple spraying robots to complete collaborative operations.

2. The method according to claim 1, characterized in that, The process of establishing a discrete state equation with a sampling period T based on the state data, designing a sliding mode switching surface by introducing a sampling period compensation term, constructing a discrete Lyapunov function, and generating a first control quantity includes: Taylor expansion of the robot dynamics equations at the sampling time is performed, and second-order terms are retained to establish discrete state equations containing the sampling period T. Based on the discrete state equation, a sliding mode switching surface with the sampling period compensation term is designed to construct a quadratic discrete Lyapunov function. By analyzing the forward difference of the discrete Lyapunov function, the stability condition is obtained; Based on the stability condition, a discrete sliding mode controller containing an equivalent control term and a switching control term is designed to generate the first control quantity.

3. The method according to claim 2, characterized in that, Design a discrete sliding mode controller that includes an equivalent control term and a switching control term, including: Based on the stability conditions, an equivalent control term is designed to compensate for the known dynamics of the system, and a switching control term is designed to counteract external disturbances. The discrete sliding mode controller employs a continuous reaching law, setting the boundary layer thickness to a value within the range of 0.01-0.

1. The first control quantity is generated by replacing the sign function with a saturation function in the switching control item.

4. The method according to claim 1, characterized in that, Constructing an adaptive barrier function includes: Set the robot joint angle limit to ±170 degrees, the speed limit to ±180 degrees / second, and the acceleration limit to ±900 degrees / square second; Introduce an auxiliary state variable ξ with a value range of 0.1-10, and construct a logarithmic barrier function B(x,ξ)=-ln((g(x)+ξ) / ξ), where g(x) represents the constraint function; By adjusting the value of the auxiliary state variable ξ, the steepness of the barrier function is dynamically changed, thereby generating an adaptive barrier function.

5. The method according to claim 4, characterized in that, The barrier function parameters are optimized using gradient descent, including: Construct a composite Lyapunov function V(x,ξ) and design an adaptive law ξ'=-γ(B(x,ξ)-α), where γ is the adaptive gain with a value range of 1-10 and α is the desired barrier function value; The parameters are updated using stochastic gradient descent, with an update step size of 0.01–0.

1. The second control variable is generated by introducing a momentum coefficient of 0.9 and an adaptive learning rate strategy.

6. The method according to claim 1, characterized in that, Construct a control model for the T-product dynamic system, including: The joint angle data, angular velocity data, motor torque data, and end-effector pose data of the painting robot within the sampling period T are collected as input data. The system state space is divided into N subspaces, where N ranges from 4 to 8, and each subspace corresponds to a local linear model. We select cubic B-spline basis functions as basis functions, construct the tensor representation of the state transition matrix, and generate the T-product dynamic system model structure.

7. The method according to claim 6, characterized in that, Enables online identification and updating of model parameters, including: Based on the T-product dynamic system model structure, the Tucker decomposition method is used to decompose the high-dimensional parameter identification problem into three-dimensional sub-problems. A weighted least squares optimization problem is constructed, with weight coefficients set to 0.1-1.0, and the L2 regularization method is used to solve for the parameters of each local model. The model parameters are updated using a recursive least squares algorithm, with the exponential forgetting factor set to 0.95-0.

99. The model structure is adjusted based on a sliding time window of 100-500 sampling points, with the parameter change rate limited to within 20%, to generate the third control variable.

8. The method according to claim 1, characterized in that, A hierarchical control strategy is adopted to achieve collaborative control of multiple robots, including: Based on the third control variable, an improved Hungarian algorithm is used for task allocation. This improved Hungarian algorithm is enhanced by introducing a multi-objective optimization mechanism into the traditional Hungarian algorithm. This multi-objective optimization mechanism simultaneously considers three key factors: energy minimization, load balancing, and spraying quality. The optimization objectives include: Minimize overall energy consumption, ensure that the workload difference between each robot is less than 15%, and control the coating overlap rate to 5%-10%; An improved RRT* algorithm is used at the global layer to generate collision-free paths, with the sampling point spacing set to 50-100 mm. The improved RRT* algorithm improves upon the traditional RRT* algorithm by introducing a heuristic sampling strategy and a dynamic search radius adjustment mechanism. The heuristic sampling strategy adjusts the sampling point distribution according to the workpiece surface features and spraying requirements, while the dynamic search radius adjustment mechanism adjusts the search radius according to the environmental complexity. The trajectory is optimized using a fifth-order polynomial function at the local layer, with the position error limited to ±0.1 mm and the attitude error limited to ±0.1 degrees.

9. The method according to claim 8, characterized in that, To achieve collaborative control of multiple robots, including: A hierarchical control architecture is adopted, with a control cycle of 100 milliseconds for the upper-layer collaborative controller and 1 millisecond for the lower-layer independent controller. The coupling relationship between robots is established using the virtual structure method, with the coupling stiffness coefficient set to 10-100; Design an event-triggered distributed control strategy, with the trigger threshold set to a position error of 0.5 mm or an attitude error of 0.5 degrees; A prediction compensation mechanism of 100-200 milliseconds is introduced to ensure the stability of the system when the communication latency and data loss rate do not exceed 5%.

10. An intelligent multi-task collaborative operation system for spray painting robots, characterized in that, include: The data acquisition module is used to collect joint angles, angular velocities, torque data, and control parameters of the painting robot; The discretized sliding mode control module is used to receive data from the data acquisition module and generate a first control quantity. The discretized sliding mode control module is connected to the adaptive barrier function module through a data bus. An adaptive barrier function module is used to receive the first control quantity and generate a second control quantity in combination with robot system constraints. The adaptive barrier function module is connected to the dynamic model module through a real-time data interface. The dynamic model module is used to receive the second control quantity, perform online parameter identification, and generate the third control quantity. The dynamic model module is connected to the cooperative control module via Ethernet. The collaborative control module is used to receive the third control quantity, plan the collaborative trajectory of multiple robots, and output control commands. The collaborative control module is connected to each robot controller through a CAN bus to complete the multi-task collaborative operation of the painting robot.

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