A visual robot arm control method for asymmetric actuator backlash

Abstract

The application discloses a visual robot arm control method for asymmetric actuator backlash, comprising the following steps: establishing a coordinate transformation relationship according to a robot kinematics method; establishing a feature point projection relationship according to a perspective projection principle; constructing a robot arm dynamics model according to a Lagrange dynamics method; establishing an asymmetric actuator dynamics model with backlash according to a dynamics equation; establishing a dynamics inverse model of the actuator according to the actuator dynamics model; establishing a parameterized error and an unparameterized error according to the error between the models; establishing a new robot arm dynamics model according to the parameterized error and the unparameterized error; designing a visual servo dynamics controller with an adaptive robust compensator according to the robot arm dynamics model with the actuator backlash, and constructing a camera parameter estimation method; and constructing an inverse model adaptive parameter and a robust compensator parameter estimation method according to the robot arm dynamics model with the actuator backlash.

Description

Technical Field

[0001] This invention relates to the field of robotic arm technology, and in particular to a vision-based robotic arm control method for asymmetric actuator backlash. Background Technology

[0002] With the development of modern technology, intelligent control technology in the field of robotics has been rapidly promoted and developed. Among them, image-based visual servo control robotic arms, due to their high versatility and strong robustness, are a robot control solution with strong application value. Therefore, they have always been a key research area in both industry and academia.

[0003] With the increasing demand for robots in modern society, the requirements for robot performance are also rising, making robot control technology a focus of research. Existing robot control methods targeting actuator backlash primarily focus on symmetrical actuators. Previous research has successfully achieved inverse compensation for unknown backlash in the joint symmetrical actuators of a vision-servo controlled robotic arm. However, actuators in reality are often asymmetrical, and considering only the backlash of symmetrical joint actuators is insufficient. When performing high-precision control tasks in complex and variable environments, unknown asymmetrical backlash often affects the stability and safety of the system. Furthermore, previous control compensation schemes for symmetrical actuator backlash are inapplicable to asymmetrical actuators. Neglecting asymmetrical actuator backlash compensation schemes or using symmetrical actuator compensation schemes on asymmetrical actuators in robotic arm joints can directly lead to system instability, preventing the robotic arm from successfully completing control tasks, and even further damaging the robotic arm, causing greater losses. Therefore, in response to the above problems, this patent proposes a vision-based robotic arm control method for asymmetric actuator compensation, which aims to solve the adverse effects of unknown symmetric actuator backlash on the system, ensure the steady-state performance of the system, and realize the tracking of the vision-based robotic arm on the image position. Summary of the Invention

[0004] To address the problems existing in the prior art, the present invention aims to provide a vision-based robotic arm control method for asymmetric actuator backlash. Based on an adaptive control strategy, and using the inverse model of the asymmetric actuator backlash, a compensation technique for unknown actuator backlash based on actuator input / output is developed, along with a vision servo robotic arm control method for non-calibrated cameras. A parameterization method is used to process the unknown asymmetric actuator backlash and the unknown internal and external projection matrices of the monocular camera. An adaptive estimation method is used to estimate and compensate for the actuator parameters and the intrinsic and extrinsic parameters of the monocular camera, ensuring that in closed-loop dynamic control, the robotic arm's end effector can reach the desired position on the image through joint state feedback and visual feedback from the non-calibrated camera. This method solves the challenge of asymmetric actuators in non-calibrated vision servo control robotic arms by introducing an adaptive online update of parameter estimates. This provides a new approach and a new method for high-precision control of vision-based robotic arms.

[0005] To solve the above problems, the present invention adopts the following technical solution. A vision-based robotic arm control method for addressing backlash in asymmetric actuators, characterized by comprising the following steps: S1. Coordinate transformation relationships constructed based on robot kinematics methods; S2. Based on the principle of perspective projection, establish the projection relationship of feature points; S3. Construct a dynamic model of the robotic arm based on the Lagrange dynamics method; S4. Based on the dynamic equations, establish a dynamic model of the asymmetric actuator with backlash; S5. Based on the actuator dynamics model, establish the inverse dynamics model of the actuator; S6. Based on the error between the two models, establish the parameterized error and the non-parameterized error; S7. Establish a new dynamic model of the robotic arm based on the parameterized error and the non-parameterized error; S8. Based on the dynamic model of the robotic arm with actuator backlash, design a vision servo dynamic controller with an adaptive robust compensator; S9. Based on the dynamic model of the robotic arm with actuator backlash, construct a method for estimating camera parameters; S10. Based on the dynamic model of the robotic arm with actuator backlash, construct estimation methods for the adaptive parameters of the inverse model and the parameters of the robust compensator.

[0006] Based on robot kinematics methods, coordinate transformation relationships are constructed. Among them, using This represents the angle of each joint of the robotic arm. The three-dimensional spatial position of the feature points at the joint ends corresponds to It is the angular velocity of the joint. It is the spatial velocity of the feature point. It is the Jacobian mapping matrix from joint velocity to feature point velocity.

[0007] Based on the principle of perspective projection, establish the projection relationship of feature points. Perspective projection model using a monocular camera in, This represents the projected coordinates of the feature points on the image, with the corresponding unit being pixels; while It is the extrinsic parameter matrix of the camera. It is a rotation matrix. It is a translation vector; and That is the intrinsic parameter matrix corresponding to the camera. , It corresponds to the camera center. and It is the scaling factor along the two axes of the image plane. It is the angle between the two axes of the image plane. This refers to the depth of the projection point relative to the camera center in the projection coordinate system. Integrating the intrinsic and extrinsic parameters yields the result. Furthermore, it can be seen that the depth information of feature points relative to the camera can be written as... Differentiating the integrated projection model yields the image velocity relationship of the feature points. In non-calibrated cameras, due to the unknown parameters, the parameter matrix... It is unknown, but since the position of the projection plane can be arbitrary, in order to ensure that the parameter matrix has a unique solution, we fix the last element of the parameter matrix to a fixed value and arrange the other elements into a vector. .

[0008] Based on the Lagrange dynamics method, a dynamic model of the robotic arm is constructed: in and These are the inertial force matrix and the centripetal force Coriolis matrix of the robotic arm. It is the force of gravity acting on the robotic arm. It refers to the torque of each joint of the robotic arm.

[0009] Based on the dynamic equations, a dynamic model of an asymmetric actuator with backlash is established: Establish an input-output dynamic model for an asymmetric actuator with backlash: in and It is the system input / output. and It refers to the left and right output gain of the actuator. and It is the width of the dead zone of the left and right teeth of the actuator. When the output is displayed, it means that the actuator has retained the output from the previous time.

[0010] Based on the actuator dynamics model, establish the actuator's inverse dynamics model. in As the input to the inverse model, and It is a self-defined smooth function. in For custom gain, and it can be proven that when When the value of is large enough, the error between the inverse model and the forward model is bounded.

[0011] Based on the error between the two models, parameterized error and non-parameterized error are established. Assuming the ratio of the left and right gains of the asymmetric actuator It is known, and assume The actuator inverse model can be linearized into the following form The parameter part , The input and output vectors are .

[0012] A new dynamic model of the robotic arm is established based on the parameterized error and the non-parameterized error. Since the actuator parameters are unknown, they need to be estimated, using symbols. This represents an estimate of the unknown parameter, expressed as: Therefore, the expected actuator output is expressed as To facilitate the representation of the error between the inverse model and the forward model, we define the following parameterized error notation. Therefore, the compensation error is expressed as... in, Indicates parameterization error. This indicates the unparameterized error. When... When the value is large enough, we can obtain in, This is the upper bound of the unparalleled error. Therefore, a dynamic model of the robotic arm with actuator backlash is established. in, and This represents the parametric and non-parametric errors of each joint. .

[0013] Based on the dynamic model of a robotic arm with actuator backlash, a vision servo dynamic controller with an adaptive robust compensator is designed: To ensure that the end effector feature points of the robotic arm reach the desired positions on the image, we established a visual feedback error based on image pixels. Based on the dynamic model of the robotic arm, a dynamic controller is established to drive the robotic arm to reach the unknown desired image. in It is the real-time gravity of the robotic arm. and It is a skew-symmetric gain matrix. It is the Jacobian mapping matrix. and It is composed of the corresponding camera parameter estimates. It is a constructed adaptive robust compensator in The upper bound vector of the unparalleled error The adaptive estimate, This is the upper bound of the unparalleled error of each joint actuator. It is a symbolic function based on the change in joint angular velocity.

[0014] Based on the dynamic model of a robotic arm with actuator backlash, a method for estimating camera parameters is constructed. By combining the dynamics controller and the dynamics equations of the robotic arm with actuator backlash. Based on the above formula, a regression matrix with camera parameter estimation error is constructed. When the robotic arm is moving, it has the following estimation error. in, It is a regression matrix. The robotic arm performs non-linear motion offline. By selecting any five or more non-collinear points, an adaptive parameter update method for the uncalibrated camera is constructed. in, It is the adaptive update rate gain matrix; Based on the dynamic model of a robotic arm with actuator backlash, a method for estimating the adaptive parameters of the inverse model and the parameters of the robust compensator is constructed. By utilizing the real-time status feedback of the robotic arm, we construct the adaptive law for the inverse model parameters of each joint asymmetric actuator in the following form. in The gain matrix is ​​adaptively estimated for each joint. For the upper bound of the unparalleled error, the following adaptive update method is constructed. in, It is also an adaptive gain update; Based on the dynamic model of the vision-based robotic arm with actuator backlash, the Lyapunov function is selected: use and express and The elements are selected from the following Lyapunov functions. Scalarization of the closed-loop control dynamics equations Processing the adaptive law of camera parameters Substituting the Jacobian mapping relation, we obtain the velocity of the feature point depth. Furthermore, establish the following relationships By taking the derivative of the Lyapunov function and substituting it into the above equation, we can obtain... Selecting some items can yield By combining the adaptive update law of actuator parameters and the robust update rate of parameters, further derivation is made. Therefore, it can be deduced that According to the above inequality, the boundedness of the Lyapunov function of the system can be obtained. When time approaches infinity, the system tends to be stable and the joint angular velocity of the robotic arm decreases to zero. After the system stabilizes Simplifying one part yields Further derivation yields the following results: as well as Combining the two conditions yields .

[0015] Beneficial effects of the present invention Compared with the prior art, the advantages of this invention are: When the joints of the visual servo robotic arm are asymmetric actuators with backlash, this invention can ensure the high-precision control performance of the robotic arm.

[0016] This invention proposes an adaptive algorithm that can update the backlash inverse model parameters of an asymmetric actuator online, adjusting the parameters of the inverse model according to the real-time state of each joint.

[0017] This invention proposes a dynamic controller for an uncalibrated vision robotic arm with adaptive robust compensation, which ensures that backlash can be eliminated under uncalibrated vision, and the system error converges to zero. Attached Figure Description

[0018] Figure 1 This is a flowchart illustrating the principle of the present invention. Detailed Implementation

[0019] The technical solutions of the present invention will be clearly and completely described below with reference to the accompanying drawings of the embodiments of the present invention. Obviously, the described embodiments are only some embodiments of the present invention, and not all embodiments. All other embodiments obtained by those skilled in the art based on the embodiments of the present invention without creative effort are within the scope of protection of the present invention.

[0020] Please see Figure 1 A vision-based robotic arm control method for asymmetric actuator backlash includes the following steps: S1. Coordinate transformation relationships constructed based on robot kinematics methods; S2. Based on the principle of perspective projection, establish the projection relationship of feature points; S3. Construct a dynamic model of the robotic arm based on the Lagrange dynamics method; S4. Based on the dynamic equations, establish a dynamic model of the asymmetric actuator with backlash; S5. Based on the actuator dynamics model, establish the inverse dynamics model of the actuator; S6. Based on the error between the two models, establish the parameterized error and the non-parameterized error; S7. Establish a new dynamic model of the robotic arm based on the parameterized error and the non-parameterized error; S8. Based on the dynamic model of the robotic arm with actuator backlash, design a vision servo dynamic controller with an adaptive robust compensator; S9. Based on the dynamic model of the robotic arm with actuator backlash, construct a method for estimating camera parameters; S10. Based on the dynamic model of the robotic arm with actuator backlash, construct estimation methods for the adaptive parameters of the inverse model and the parameters of the robust compensator.

[0021] Based on robot kinematics methods, coordinate transformation relationships are constructed. Among them, using This represents the angle of each joint of the robotic arm. The three-dimensional spatial position of the feature points at the joint ends corresponds to It is the angular velocity of the joint. It is the spatial velocity of the feature point. It is the Jacobian mapping matrix from joint velocity to feature point velocity.

[0022] Based on the principle of perspective projection, establish the projection relationship of feature points. Perspective projection model using a monocular camera in, This represents the projected coordinates of the feature points on the image, with the corresponding unit being pixels; while It is the extrinsic parameter matrix of the camera. It is a rotation matrix. It is a translation vector; and That is the intrinsic parameter matrix corresponding to the camera. , It corresponds to the camera center. and It is the scaling factor along the two axes of the image plane. It is the angle between the two axes of the image plane. This refers to the depth of the projection point relative to the camera center in the projection coordinate system. Integrating the intrinsic and extrinsic parameters yields the result. Furthermore, it can be seen that the depth information of feature points relative to the camera can be written as... Taking the derivative of the integrated projection model yields the image velocity relationship of the feature points. In non-calibrated cameras, due to the unknown parameters, the parameter matrix... It is unknown, but since the position of the projection plane can be arbitrary, in order to ensure that the parameter matrix has a unique solution, we fix the last element of the parameter matrix to a fixed value and arrange the other elements into a vector. .

[0023] Based on the Lagrange dynamics method, a dynamic model of the robotic arm is constructed: in and These are the inertial force matrix and the centripetal force Coriolis matrix of the robotic arm. It is the force of gravity acting on the robotic arm. It refers to the torque of each joint of the robotic arm.

[0024] Based on the dynamic equations, a dynamic model of an asymmetric actuator with backlash is established: Establish an input-output dynamic model for an asymmetric actuator with backlash: in and It is the system input / output. and It refers to the left and right output gain of the actuator. and It is the width of the dead zone of the left and right teeth of the actuator. When the output is not displayed, it means that the actuator has retained the output from the previous time.

[0025] Based on the actuator dynamics model, establish the actuator's inverse dynamics model. in As the input to the inverse model, and It is a self-defined smooth function. in For custom gain, and it can be proven that when When the value of is large enough, the error between the inverse model and the forward model is bounded.

[0026] Based on the error between the two models, parameterized error and non-parameterized error are established. Assuming the ratio of the left and right gains of the asymmetric actuator It is known, and assume The actuator inverse model can be linearized into the following form The parameter part , The input and output vectors are .

[0027] A new dynamic model of the robotic arm is established based on the parameterized error and the non-parameterized error. Since the actuator parameters are unknown, they need to be estimated, using symbols. This represents an estimate of the unknown parameter, expressed as: Therefore, the expected actuator output is expressed as To facilitate the representation of the error between the inverse model and the forward model, we define the following parameterized error notation. Therefore, the compensation error is expressed as... in, Indicates parameterization error. This indicates the unparameterized error. When... When the value is large enough, we can obtain in, This is the upper bound of the unparalleled error. Therefore, a dynamic model of the robotic arm with actuator backlash is established. in, and This represents the parametric and non-parametric errors of each joint. .

[0028] Based on the dynamic model of a robotic arm with actuator backlash, a vision servo dynamic controller with an adaptive robust compensator is designed: To ensure that the end effector feature points of the robotic arm reach the desired positions on the image, we established a visual feedback error based on image pixels. Based on the dynamic model of the robotic arm, a dynamic controller is established to drive the robotic arm to reach the unknown desired image. in It is the real-time gravity of the robotic arm. and It is a skew-symmetric gain matrix. It is the Jacobian mapping matrix. and It is composed of the corresponding camera parameter estimates. It is a constructed adaptive robust compensator in The upper bound vector of the unparalleled error The adaptive estimate, This is the upper bound of the unparalleled error of each joint actuator. It is a symbolic function based on the change in joint angular velocity.

[0029] Based on the dynamic model of a robotic arm with actuator backlash, a method for estimating camera parameters is constructed. By combining the dynamics controller and the dynamics equations of the robotic arm with actuator backlash. Based on the above formula, a regression matrix with camera parameter estimation error is constructed. When the robotic arm is moving, it has the following estimation error. in, It is a regression matrix. The robotic arm performs non-linear motion offline. By selecting any five or more non-collinear points, an adaptive parameter update method for the uncalibrated camera is constructed. in, It is the adaptive update rate gain matrix; Based on the dynamic model of a robotic arm with actuator backlash, a method for estimating the adaptive parameters of the inverse model and the parameters of the robust compensator is constructed. By utilizing the real-time status feedback of the robotic arm, we construct the adaptive law for the inverse model parameters of each joint asymmetric actuator in the following form. in The gain matrix is ​​adaptively estimated for each joint. For the upper bound of the unparalleled error, the following adaptive update method is constructed. in, It is also an adaptive gain update; Based on the dynamic model of the vision-based robotic arm with actuator backlash, the Lyapunov function is selected: use and express and The elements are selected from the following Lyapunov functions. Scalarization of the closed-loop control dynamics equations Processing the adaptive law of camera parameters Substituting the Jacobian mapping relation, we obtain the velocity of feature point depth. Furthermore, establish the following relationships By taking the derivative of the Lyapunov function and substituting it into the above equation, we can obtain... Selecting some items can yield By combining the adaptive update law of actuator parameters and the robust update rate of parameters, further derivation is made. Therefore, it can be deduced that According to the above inequality, the boundedness of the Lyapunov function of the system can be obtained. When time approaches infinity, the system tends to be stable and the joint angular velocity of the robotic arm decreases to zero. After the system stabilizes Simplifying one part yields Further derivation yields the following results: as well as Combining the two conditions yields .

[0030] Based on the above derivation, it can be seen that all control signals in the closed-loop system will converge to the desired steady-state value, and the end effector position of the robotic arm under closed-loop control will reach the desired position on the image. Adaptive estimation is performed for unknown camera parameters, actuator parameters, and the upper bound of the model error, based on an inverse compensation controller. A vision-based robotic arm with asymmetric joint actuator backlash can be driven by feedback from the robotic arm's state and from a non-calibrated vision camera. The robot arm's end effector reaches the desired position without static error, demonstrating the feasibility of the robot system control scheme in this patent.

[0031] The above description is merely a preferred embodiment of the present invention; however, the scope of protection of the present invention is not limited thereto. Any equivalent substitutions or modifications made by those skilled in the art within the scope of the technology disclosed in the present invention, based on the technical solution and its improved concepts, should be covered within the scope of protection of the present invention.

Claims

1. A vision-based robotic arm control method for asymmetric actuator backlash, characterized in that, Includes the following steps: S1. Based on robot kinematics methods, construct coordinate transformation relationships; S2. Based on the principle of perspective projection, establish the projection relationship of feature points; S3. Construct a dynamic model of the robotic arm based on the Lagrange dynamics method; S4. Based on the dynamic equations, establish a dynamic model of the asymmetric actuator with backlash; S5. Based on the actuator dynamics model, establish the inverse dynamics model of the actuator; S6. Based on the error between the two models, establish the parameterized error and the non-parameterized error; S7. Establish a new dynamic model of the robotic arm based on the parameterized error and the non-parameterized error; S8. Based on the dynamic model of the robotic arm with actuator backlash, design a vision servo dynamic controller with an adaptive robust compensator; S9. Based on the dynamic model of the robotic arm with actuator backlash, construct a method for estimating camera parameters; S10. Based on the dynamic model of the robotic arm with actuator backlash, construct an estimation method for the adaptive parameters of the inverse model and the parameters of the robust compensator; Based on the dynamic equations, a dynamic model of an asymmetric actuator with backlash is established: Establish an input-output dynamic model for an asymmetric actuator with backlash: in and It is the system input / output. and It refers to the left and right output gain of the actuator. and It is the width of the dead zone of the left and right teeth of the actuator. When output is displayed, it means that the actuator has retained the output from the previous time step. Based on the actuator dynamics model, establish the actuator's inverse dynamics model. in As the input to the inverse model, and It is a self-defined smooth function. in For custom gain, and it can be proven that when When the value of is large enough, the error between the inverse model and the forward model is bounded.

2. The vision-based robotic arm control method for asymmetric actuator backlash according to claim 1, characterized in that: Based on robot kinematics methods, coordinate transformation relationships are constructed. Among them, using This represents the angle of each joint of the robotic arm. The three-dimensional spatial position of the feature points at the joint ends corresponds to It is the angular velocity of the joint. It is the spatial velocity of the feature point. It is the Jacobian mapping matrix from joint velocity to feature point velocity.

3. The vision-based robotic arm control method for asymmetric actuator backlash according to claim 2, characterized in that: Based on the principle of perspective projection, establish the projection relationship of feature points. Perspective projection model using a monocular camera in, This represents the projected coordinates of the feature points on the image, with the corresponding unit being pixels; while It is the extrinsic parameter matrix of the camera. It is a rotation matrix. It is a translation vector; and That is the intrinsic parameter matrix corresponding to the camera. , It corresponds to the camera center. and It is the scaling factor along the two axes of the image plane. It is the angle between the two axes of the image plane. This refers to the depth of the projection point relative to the camera center in the projection coordinate system. Integrating the intrinsic and extrinsic parameters yields the result. Furthermore, it can be seen that the depth information of feature points relative to the camera can be written as... Taking the derivative of the integrated projection model yields the image velocity relationship of the feature points. In non-calibrated cameras, due to the unknown parameters, the parameter matrix... It is unknown, but since the position of the projection plane can be arbitrary, in order to ensure that the parameter matrix has a unique solution, we fix the last element of the parameter matrix to a fixed value and arrange the other elements into a vector. 。 4. The vision-based robotic arm control method for asymmetric actuator backlash according to claim 3, characterized in that: Based on the Lagrange dynamics method, a dynamic model of the robotic arm is constructed: in and These are the inertial force matrix and the centripetal force Coriolis matrix of the robotic arm. It is the force of gravity acting on the robotic arm. It refers to the torque of each joint of the robotic arm.

5. A vision-based robotic arm control method for asymmetric actuator backlash according to claim 4, characterized in that: Based on the error between the two models, parameterized error and non-parameterized error are established. Assuming the ratio of the left and right gains of the asymmetric actuator It is known, and assume The inverse model of the actuator can be linearized into the following form: The parameter part , The input and output vectors are .

6. The vision-based robotic arm control method for asymmetric actuator backlash according to claim 5, characterized in that: A new dynamic model of the robotic arm is established based on the parameterized error and the non-parameterized error. Since the actuator parameters are unknown, they need to be estimated, using symbols. This represents an estimate of the unknown parameter, expressed as: Therefore, the expected actuator output is expressed as To facilitate the representation of the error between the inverse model and the forward model, we define the following parameterized error notation. Therefore, the compensation error is expressed as... in, Indicates parameterization error. Indicates unparameterized error, when When the value is large enough, we can obtain in, This is the upper bound of the unparameterized error; therefore, a dynamic model of the robotic arm with actuator backlash is established. in, and This represents the parametric and non-parametric errors of each joint. 。 7. A vision-based robotic arm control method for asymmetric actuator backlash according to claim 6, characterized in that: Based on the dynamic model of a robotic arm with actuator backlash, a vision servo dynamic controller with an adaptive robust compensator is designed: To ensure that the end effector feature points of the robotic arm reach the desired positions on the image, we established a visual feedback error based on image pixels. Based on the dynamic model of the robotic arm, a dynamic controller is established to drive the robotic arm to reach the unknown desired image. in It is the real-time gravity of the robotic arm. and It is a skew-symmetric gain matrix. It is the Jacobian mapping matrix. and It consists of the corresponding estimated camera parameters. It is a constructed adaptive robust compensator in The upper bound vector of the unparalleled error The adaptive estimate, This is the upper bound of the unparalleled error of each joint actuator. It is a symbolic function based on the change in joint angular velocity.

8. A vision-based robotic arm control method for asymmetric actuator backlash according to claim 7, characterized in that: Based on the dynamic model of a robotic arm with actuator backlash, a method for estimating camera parameters is constructed. By combining the dynamics controller and the dynamics equations of the robotic arm with actuator backlash. Based on the above formula, a regression matrix with camera parameter estimation error is constructed. When the robotic arm is moving, it has the following estimation error. in, It is a regression matrix. The robotic arm performs non-linear motion offline. By selecting any five or more non-collinear points, an adaptive parameter update method for the uncalibrated camera is constructed. in, It is the adaptive update rate gain matrix; Based on the dynamic model of a robotic arm with actuator backlash, a method for estimating the adaptive parameters of the inverse model and the parameters of the robust compensator is constructed. By utilizing the real-time status feedback of the robotic arm, we construct the adaptive law for the inverse model parameters of each joint asymmetric actuator in the following form. in The gain matrix is ​​adaptively estimated for each joint, and an adaptive update method is constructed to address the upper bound of the unparalleled error. in, It is also an adaptive gain update; Based on the dynamic model of the vision-based robotic arm with actuator backlash, the Lyapunov function is selected: use and express and The elements are selected from the following Lyapunov functions. Scalarization of the closed-loop control dynamics equations Processing the adaptive law of camera parameters Substituting the Jacobian mapping relation, we obtain the velocity of feature point depth. Furthermore, establish the following relationships By taking the derivative of the Lyapunov function and substituting it into the above equation, we can obtain... Selecting some items can yield By combining the adaptive update law of actuator parameters and the robust update rate of parameters, further derivation is made. Therefore, it can be deduced that According to the above inequality, the boundedness of the Lyapunov function of the system can be obtained. When time approaches infinity, the system tends to be stable and the joint angular velocity of the robotic arm decreases to zero. After the system stabilizes Simplifying one part yields Further derivation yields the following results: as well as Combining the two conditions yields 。

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