Abstract: Based on the research of calibration-free visual servoing methods for robots based on active disturbance rejection controllers (ADRCs), a novel dual-loop structure ARC-free visual servoing control method for robots is proposed. The inner loop uses the Kalman filtering algorithm for online identification of the image Jacobian matrix, which can approximate the real model well. The outer loop uses an ADRC, which uses a nonlinear observer to estimate the total disturbance of the system relative to the current estimated model in real time and dynamically compensates for it in the control. A two-dimensional moving target tracking experiment was conducted on a six-degree-of-freedom industrial robot. The experimental results show the feasibility and effectiveness of the proposed method. Keywords: Kalman filtering; active disturbance rejection controller; calibration-free; visual servoing 1 Introduction In recent years, calibration-free robot visual servoing control has gradually become a research hotspot in the field of robot visual servoing control due to its flexibility and robustness. Calibration-free refers to designing control laws directly using system image information to drive robot motion and achieve operation on moving targets without knowing the hand-eye relationship model. To solve this problem, it is necessary to know the nonlinear mapping relationship between the robot's hand and eye. According to the different models used to describe this nonlinear relationship, the existing calibration-free methods can be divided into two categories. One type is based on the image Jacobian matrix, which is the most basic method for uncalibrated robot vision control. Its basic idea is to estimate the nonlinear mapping relationship between the robot's hand and eye in real time using a linear image Jacobian matrix at each moment, and then derive the motion control quantity for the next moment. The second type uses neural networks to fit the nonlinear mapping relationship between the robot's hand and eye. This method requires pre-training with a large number of sample points in the robot's motion space offline, which is sometimes difficult to obtain for multi-degree-of-freedom motion spaces. Su Jianbo et al. applied model-free theory and the concept of Auto Disturbance Rejection Controller (ADRC) to the field of uncalibrated robot vision servoing, designing a hand-eye coordination controller that is not dependent on a specific task and achieves good results. However, the disturbance compensation accuracy of ADRC is related to the difference between the estimated system model and the actual system model; the smaller the difference, the higher the accuracy; the larger the difference, the lower the accuracy, and it may even lead to system instability. Based on the above reasons, this paper combines the online image Jacobian matrix identification method with the ADRC-based robot calibration-free visual servoing method, and proposes a new double-loop structure robot calibration-free active disturbance rejection visual servoing control method. 2 Problem Description The basic idea of ​​ADRC-based robot calibration-free visual servoing control is to first estimate the initial value of the system's image Jacobian matrix, and use it as an approximate linear model of the system, which remains unchanged during the motion. The nonlinear model error caused by this approximation (i.e., system internal disturbance) and the sum of image detection error, unknown external disturbance, etc. (i.e., system external disturbance) are regarded as the total disturbance of the system. A nonlinear state observer (Extended State Observer, ESO) is used to estimate it in real time and dynamically compensate it in the control. The structure of the whole system is shown in Figure 1. In the figure, () represents the desired position coordinates of the moving target, which can be obtained by the first-order prediction as shown in Equation (1): Where, represents. (k) represents the position vector of the target in the image coordinate system at time k. Let represent the position of the robot's hand in the image coordinate system at time k, represent the position of the target in the spatial coordinate system, Zx1 and Zx2 are the outputs of the extended state observer (ESO) in the x direction, Zy1 and Zy2 are the outputs of the extended state observer (ESO) in the y direction, Uox and Uoy are the outputs of the nonlinear state error feedback controller (NLSEF) in the x and y directions, and represent the system disturbance. The image Jacobian matrix J is used to describe the nonlinear mapping relationship between the robot's hand and eye, where J is defined as shown in equation (2). In the equation, is the motion velocity of the robot's hand in the robot's motion space, which is the motion control quantity u of the robot, i.e., its motion velocity in the image feature space. After discretization, it is: The initial estimate of this matrix can be obtained through the initial trial motion and remains unchanged throughout the control process. Experiments show that when the variation range of J is small, this control strategy has a certain robustness. However, when the object model changes significantly, the overall control effect of the system deteriorates. If the controlled object has a known part, this part can be compensated for in the input of the extended state observer, and only the uncertain part can be estimated, thus improving the estimation accuracy of the extended state observer. In this system, when the target is stationary, the variation range of J is small, and the extended state observer can better estimate the real-time disturbance of the system, resulting in good control performance. When the target moves, the variation range of the actual image Jacobian matrix increases. The faster the target moves, the more obvious the change in the image Jacobian matrix becomes. As shown in Figure 2, Figure 2(a) shows the trajectory of the target and the hand on the image plane, and Figure 2(b) shows the variation curve of the corresponding image Jacobian matrix values ​​(taking J[sub]11[/sub](k) and J[sub]22[/sub](k) as examples). If the image Jacobian matrix value estimated by the initial trial motion is still used as the system model, the difference between the estimated model and the actual system model will be too large, reducing the estimation accuracy of the extended state observer used to estimate the real-time disturbance of the system, thereby causing a decrease in the control performance of the entire system. 3. Calibration-Free Active Disturbance Rejection (ADRC) Vision Servo Control System for Robots Based on a Dual-Loop Structure Servo Control Principle: As the above analysis shows, if we can accurately estimate the value of the image Jacobian matrix at each moment, instead of using a fixed image Jacobian matrix value to approximate the real-time changing system model, we can reduce the difference between the estimated model and the actual model, improve the estimation accuracy of the extended state observer, and thus enable it to better estimate the total disturbance of the system relative to the current estimated model in real time, improve compensation accuracy, and ultimately achieve higher-performance target localization and tracking. Based on this, this paper combines the online image Jacobian matrix identification method with the ADRC-based calibration-free vision servoing method for robots to construct a calibration-free ADRC vision servo control system for robots with a dual-loop structure, as shown in Figure 3. The inner loop uses the Kalman filtering algorithm for online image Jacobian matrix identification, continuously estimating and updating the hand-eye relationship model at the current moment through the motion features of the previous few moments to better approximate the real model. The outer loop uses an active disturbance rejection controller. Due to the effect of the inner loop, the difference between the estimated model and the actual model is small. The nonlinear observer can better estimate the total disturbance of the system relative to the current estimated model and make dynamic compensation in the control, thereby improving the performance of the visual servo system. For a monocular fixed-plane visual servo system (u[sub]z[/sub]=0), when the target is stationary or moving at a constant speed, equation (3) can be written as: Where J[sub]11[/sub](k) and J[sub]22[/sub](k) are identified online by the Kalman filtering algorithm. 1 is the sum of the system modeling error and the unknown external disturbance, which is called the total disturbance of the system, and is estimated in real time by ESO. Compared with the total disturbance of the system directly based on ADRC, it is obvious that when the image Jacobian matrix changes significantly, the total disturbance of the system is greatly reduced. At this time, a discrete ESO and nonlinear state error feedback law (NLSEF) based on a double-ring structure can be designed as follows: where h is the sampling step size, and the expression of the nonlinear function fal() is as follows: In equation (4), the online identification of the image Jacobian matrix can be realized by Kalman filtering. The basic idea is to first construct a linear time-varying system, and use the parameters of the image Jacobian matrix to be estimated as the state of the system, that is, to transform the estimation of the image Jacobian matrix into the observation of the state of this linear time-varying system. Then, the state of this system is observed by the Kalman filtering algorithm. Since the algorithm considers the influence of noise on the observation of the system state, it has strong robustness. The algorithm can be simply described as follows: step1: Construct a linear time-varying system and define the state of the system as: step2: According to the Kahnan filtering estimation algorithm, establish a recursive estimation where is the noise variance matrix, which is set according to the actual noise situation, and P(k) is the state estimation error variance matrix. The initial value of the state estimate, x(O) (i.e. J(0)), can be obtained using the initial trial motion. Subsequently, the information obtained from the robot's completed motion can be directly used in the estimation of the Jacobian matrix, without the need to introduce redundant trial motion. 4. Experimental Study The experimental setup is shown in Figure 4. The hardware of the system mainly consists of: a MOTOMAN SV3 robot, a SONY 1/3” color CCD camera, a robot control cabinet, a control computer, and a Beijing Microvision image acquisition card. The CCD camera is fixedly installed above the workbench, allowing simultaneous observation of the robot's gripper and the target. The CCD acquires images of both the target and the robot's gripper, and after image processing and recognition, the positions of the target and the robot's gripper in the image are obtained. To simplify the image processing, the camera uses two different colored square blocks to distinguish between the gripper and the target, and the centroid coordinates of the two color blocks are taken as the image coordinates of the gripper and the target. RS-232 serial communication is used, with a sampling period of 500 ms. The experimental task is for the target to perform a polygonal motion on a given robot working plane. The robot's gripper is driven to track the target performing arbitrary polygonal motion on the robot's working plane using a traditional PI control strategy, an ARDC-based control strategy, and the proposed dual-loop structure control strategy, respectively, so that the two coincide on the image plane. Experiment 1: Uncalibrated visual servoing control of a robot based on PI controller. The PI controller parameters are set as follows: Proportional coefficient: 0.28 in the X direction and 0.3 in the Y direction; Integral coefficient: 0.0008 in the X direction and 0.0012 in the Y direction. The experimental results are shown in Figure 5, where Figure 5(a) and (b) represent the motion trajectories of the target and the gripper in the image plane and in the Y direction, respectively. Experiment 2: Uncalibrated visual servoing control of a robot based on ARDC controller. The ARDC controller parameters are set as follows: The experimental results are shown in Figure 6. Figures 6(a) and (b) show the motion trajectories of the target and the hand in the image plane and the Y direction, respectively. Figure 6(c) shows the change curve of the image Jacobian matrix during the control process (taking J[sub]11[/sub](k) and J[sub]22[/sub](k) as examples). Figure 6(d) shows the change curve of the system's internal disturbance. The experimental results show that when the hand-eye relationship parameters change small and have strong linear characteristics, both PI control and ADRC control can achieve good visual tracking effects. When the hand-eye relationship parameters change large, the tracking performance of both will decrease, but the latter has stronger tracking performance and robustness to changes in object model parameters than the former. Experiment 3: Online identification of image Jacobian matrix for robot uncalibrated self-disturbance visual servo control based on double-loop structure. The parameter settings are as follows: The initial value of the state noise variance matrix 2 is a two-dimensional identity matrix, the image observation noise variance matrix, and the initial value of the state estimation variance matrix P(k) is P(0) = 10[sup]5[/sup]I[sub]2＊2[/sub]. The initial value of the state estimation obtained through the initial trial motion is x(0) (i.e. J(0)), which is the same as in Experiment 2, and is still J. The ADRC controller parameters are set as follows: The experimental results are shown in Figure 7, where Figure 7(a) and (b) represent the motion trajectories of the target and the hand in the image plane and the Y direction, respectively. Figure 7(c) represents the change curve of the image Jacobian matrix during the control process (taking J[sub]11[/sub](k) and J[sub]22[/sub](k) as examples). Figure 7(d) represents the change curve of the system internal disturbance 口(k) when the Jacobian matrix is ​​fixed. Figure 7(e) represents the change curve of the system internal disturbance after introducing the Jacobian matrix online identification algorithm. The experimental results show that by adopting the dual-loop control method of online identification based on Kalman filtering for uncalibrated active disturbance rejection visual servoing, the internal disturbance caused by changes in the image Jacobian matrix is ​​reduced, and the control performance of the system is improved. Meanwhile, the experiments also revealed that when tracking fast-moving targets, directly using ADRC tracking performance degrades. However, simply combining the uncalibrated visual servoing method based on online identification of the image Jacobian matrix with the uncalibrated visual servoing method based on ADRC does not significantly improve system performance. This is because the dual-loop structure reduces the system's modeling error (i.e., internal disturbance). If the parameters of ESO (mainly the parameters {bx2, by2} used to estimate the real-time disturbance of the system) are not correspondingly reduced, it is equivalent to increasing {b+b}. The value of } leads to oscillations in the estimated value, which in turn reduces the estimation accuracy of ESO and consequently degrades the performance of the entire system. The exact quantitative relationship between the reduction in ESO-related parameters and the internal disturbance requires further in-depth research. However, it is certain that this relationship, independent of the specific system, exists and is nonlinear. 5. Conclusion This paper combines the online image Jacobian matrix identification method with the ADRC-based uncalibrated visual servoing method for robots, proposing a dual-loop structure for uncalibrated active disturbance rejection visual servoing control. The inner loop uses the Kalman filtering algorithm for online image Jacobian matrix identification, which can approximate the real visual mapping model well. The outer loop uses an active disturbance rejection controller, utilizing a nonlinear observer to estimate the total disturbance of the system relative to the current estimated model in real time and dynamically compensates for it during control. Tracking experiments on two-dimensional moving targets were conducted on a six-degree-of-freedom industrial robot. Experimental results show that this method, to a certain extent, solves the problem of reduced controller compensation accuracy and decreased system control performance in uncalibrated visual servoing systems based on active disturbance rejection controllers when the target performs large-range movements, due to the large difference between the estimated model and the actual model.