Serial robot end position calibration method and serial robot

Abstract

This application provides a method for end-effector pose calibration of a serial robot and a serial robot, which can be applied to the field of robot calibration technology. The method includes: in response to a triggered calibration task, determining the predicted pose of the serial robot's end-effector in the measurement coordinate system using the initial geometric parameters between adjacent joints, the forward kinematics model, and the transformation relationship between the measurement coordinate system and the base coordinate system; subtracting the predicted pose from the measured pose of the end-effector in the measurement coordinate system measured by a target sensor to determine a target residual; determining the parameter correction amount of the initial geometric parameters between adjacent joints based on the error mapping relationship between the target residual and the initial geometric parameters to the end-effector pose; correcting the initial geometric parameters based on the parameter correction amount to obtain corrected geometric parameters; and updating the forward kinematics model using the corrected geometric parameters so that the serial robot uses the updated forward kinematics model to determine the end-effector pose.

Description

Technical Field

[0001] This application relates to the field of robot calibration technology, specifically to a method for calibrating the end-effector pose of a serial robot and the serial robot itself. Background Technology

[0002] Serial robots have been widely used in industrial applications such as automobile assembly and precision machining. However, due to unavoidable geometric error factors such as link size error, component machining and assembly error, and structural deformation during the processing and assembly process, although serial robots have high repeatability positioning accuracy, their absolute positioning accuracy is still low, which is difficult to meet the accuracy requirements of high-end manufacturing inspection fields and restricts the development of industrial production.

[0003] There are two main approaches to improving the absolute accuracy of robot end effectors in related technologies: one is to improve the manufacturing precision of key components so that the structural parameters match the design parameters as closely as possible. Although this method can improve the accuracy to a certain extent, it will significantly increase the manufacturing cost and has poor economic efficiency. The other is to identify the structural parameter errors of serial robots in order to improve the absolute positioning accuracy of the robots. However, the accuracy of related technologies in identifying structural parameter errors is low, resulting in the robot's absolute positioning accuracy still being poor and low. Summary of the Invention

[0004] In view of the above problems, this application provides a method for end-effector pose calibration of a serial robot and a serial robot.

[0005] According to one aspect of this application, an end-effector pose calibration method for a serial robot is provided, comprising: in response to a triggered calibration task, determining the predicted pose of the end-effector in the measurement coordinate system using initial geometric parameters between adjacent joints of the serial robot, a forward kinematics model, and a transformation relationship between the measurement coordinate system and the base coordinate system, wherein the initial geometric parameters are used to describe the spatial pose relationship between adjacent joints; subtracting the predicted pose from the measured pose of the end-effector in the measurement coordinate system measured by a target sensor to determine a target residual; determining a parameter correction amount for the initial geometric parameters between adjacent joints based on the error mapping relationship between the target residual and the initial geometric parameters on the end-effector pose, wherein the error mapping relationship is used to characterize the influence of changes in the initial geometric parameters on the end-effector pose; correcting the initial geometric parameters based on the parameter correction amount to obtain corrected geometric parameters; in response to the corrected geometric parameters not meeting a stopping condition, using the corrected geometric parameters as the initial geometric parameters for the next round and repeating the above steps; in response to the corrected geometric parameters meeting the stopping condition, updating the forward kinematics model using the corrected geometric parameters so that the serial robot uses the updated forward kinematics model to determine the end-effector pose.

[0006] Another aspect of this application provides an end-effector pose calibration system for a serial robot, comprising: a first determining module, configured to, in response to a triggered calibration task, determine the predicted pose of the serial robot's end-effector in the measurement coordinate system using initial geometric parameters between adjacent joints of the serial robot, a positive kinematic model, and the transformation relationship between the measurement coordinate system and the base coordinate system, wherein the initial geometric parameters are used to describe the spatial pose relationship between adjacent joints; a calculation module, configured to subtract the predicted pose from the measured pose of the end-effector in the measurement coordinate system measured by a target sensor to determine a target residual; and a second determining module, configured to, based on the target residual and the initial geometric parameters, adjust the end-effector pose... The system employs several steps: an error mapping relationship for pose determination to define the parameter correction amount for the initial geometric parameters between adjacent joints, and an update module to update the forward kinematics model using the corrected geometric parameters. The first step involves determining the error mapping relationship to characterize the impact of changes in the initial geometric parameters on the end-effector pose. The second step involves determining the corrected geometric parameters based on the parameter correction amount. The third step involves determining the corrected geometric parameters in response to the stop condition, using them as the initial geometric parameters for the next round and repeating the above steps. The fourth step involves updating the forward kinematics model in response to the stop condition, enabling the serial robot to determine the end-effector pose using the updated forward kinematics model.

[0007] Another aspect of this application provides a serial robot, including: an end-effector pose calibration device; wherein the end-effector pose calibration device is used to perform the above-described end-effector pose calibration method for the serial robot.

[0008] Another aspect of this application provides an electronic device comprising: one or more processors; and a memory for storing one or more computer programs, wherein the one or more processors execute the one or more computer programs to implement the steps of the method described above.

[0009] Another aspect of this application provides a computer-readable storage medium having a computer program or instructions stored thereon, which, when executed by a processor, implement the steps of the above-described method.

[0010] Another aspect of this application provides a computer program product, including a computer program or instructions that, when executed by a processor, implement the steps of the above-described method.

[0011] According to embodiments of this application, by introducing a transformation relationship between the measurement coordinate system and the base coordinate system, the predicted end-effector pose output by the forward kinematics model is transformed into the measurement coordinate system and subtracted from the actual end-effector pose measured by the target sensor. This achieves the measurement of the absolute end-effector pose and the calculation of residuals, improving the completeness of parameter identification information. Based on the error mapping relationship between the target residual and the initial geometric parameters to the end-effector pose, the correction amount of each geometric parameter is determined, and the initial geometric parameters are iteratively corrected until the stopping condition is met. This achieves systematic iterative calibration of the geometric parameters of the serial robot, enabling the corrected geometric parameters to accurately reflect the robot's true physical structure. Therefore, this at least partially solves the technical problems of high cost or poor positioning accuracy, achieving the technical effect of improving the absolute positioning accuracy of the robot. Attached Figure Description

[0012] The above-mentioned contents, other objects, features and advantages of this application will become clearer from the following description of embodiments of this application with reference to the accompanying drawings.

[0013] Figure 1 The diagram illustrates an application scenario of the end-effector pose calibration method and system for a serial robot according to an embodiment of this application.

[0014] Figure 2 A flowchart of an end-effector pose calibration method for a serial robot according to an embodiment of this application is shown.

[0015] Figure 3 The spatial pose relationship between adjacent joints according to an embodiment of this application is shown.

[0016] Figure 4 A structural block diagram of a serial robot according to an embodiment of this application is shown.

[0017] Figure 5 A structural block diagram of an end-effector pose calibration system for a serial robot according to an embodiment of this application is shown.

[0018] Figure 6 A block diagram of an electronic device suitable for implementing an end-effector pose calibration method for a serial robot, according to an embodiment of this application, is shown. Detailed Implementation

[0019] The embodiments of this application will now be described with reference to the accompanying drawings. However, it should be understood that these descriptions are exemplary only and are not intended to limit the scope of this application. In the following detailed description, numerous specific details are set forth to provide a thorough understanding of the embodiments of this application for ease of explanation. However, it will be apparent that one or more embodiments may be implemented without these specific details. Furthermore, descriptions of well-known structures and technologies are omitted in the following description to avoid unnecessarily obscuring the concepts of this application.

[0020] The terminology used herein is for the purpose of describing particular embodiments only and is not intended to limit the scope of this application. The terms “comprising,” “including,” etc., as used herein indicate the presence of the stated features, steps, operations, and / or components, but do not exclude the presence or addition of one or more other features, steps, operations, or components.

[0021] All terms used herein (including technical and scientific terms) have the meanings commonly understood by those skilled in the art, unless otherwise defined. It should be noted that the terms used herein are to be interpreted in a manner consistent with the context of this specification, and not in an idealized or overly rigid way.

[0022] When using expressions such as "at least one of A, B and C", they should generally be interpreted in accordance with the meaning that is commonly understood by those skilled in the art (e.g., "a system having at least one of A, B and C" should include, but is not limited to, a system having A alone, a system having B alone, a system having C alone, a system having A and B, a system having A and C, a system having B and C, and / or a system having A, B and C, etc.).

[0023] Embodiments of this application provide an end-effector pose calibration method for a serial robot, comprising: in response to a triggered calibration task, determining the predicted pose of the serial robot end-effector in the measurement coordinate system using initial geometric parameters between adjacent joints of the serial robot, a forward kinematics model, and the transformation relationship between the measurement coordinate system and the base coordinate system, wherein the initial geometric parameters are used to describe the spatial pose relationship between adjacent joints; subtracting the predicted pose from the measured pose of the end-effector in the measurement coordinate system measured by a target sensor to determine a target residual; determining the parameter correction amount of the initial geometric parameters between adjacent joints based on the error mapping relationship between the target residual and the initial geometric parameters on the end-effector pose, wherein the error mapping relationship is used to characterize the influence of changes in the initial geometric parameters on the end-effector pose; correcting the initial geometric parameters based on the parameter correction amount to obtain corrected geometric parameters; in response to the corrected geometric parameters not meeting the stopping condition, using the corrected geometric parameters as the initial geometric parameters for the next round, and repeating the above steps; in response to the corrected geometric parameters meeting the stopping condition, updating the forward kinematics model using the corrected geometric parameters so that the serial robot can use the updated forward kinematics model to determine the end-effector pose.

[0024] Figure 1 The diagram illustrates an application scenario of the end-effector pose calibration method and system for a serial robot according to an embodiment of this application.

[0025] like Figure 1As shown, application scenario 100 according to this embodiment may include a first terminal device 101, a second terminal device 102, a third terminal device 103, a network 104, a server 105, and a serial robot 106. The network 104 serves as a medium for providing communication links between the first terminal device 101, the second terminal device 102, the third terminal device 103, and the server 105. The network 104 may include various connection types, such as wired or wireless communication links, or fiber optic cables, etc.

[0026] Users can use the first terminal device 101, the second terminal device 102, and the third terminal device 103 to interact with the server 105 via the network 104 to receive or send messages, etc. Various communication client applications can be installed on the first terminal device 101, the second terminal device 102, and the third terminal device 103, such as shopping applications, web browser applications, search applications, instant messaging tools, email clients, social media platform software, etc. (for example only).

[0027] The first terminal device 101, the second terminal device 102, and the third terminal device 103 can be various electronic devices with displays and support web browsing, including but not limited to smartphones, tablets, laptops, and desktop computers.

[0028] Server 105 can be a server that provides various services, such as a background management server that supports calibration tasks sent by users using the first terminal device 101, the second terminal device 102, and the third terminal device 103 (this is just an example). The background management server can analyze and process received user requests, such as calibration tasks, and feed back the processing results (such as correcting geometric parameters) to the terminal devices.

[0029] The serial robot 106 can be a serial multi-axis industrial robot, including but not limited to a six-axis serial industrial robot, a collaborative robot, or other open-chain mechanical structure consisting of multiple rotary joints and / or translational joints connected in series. The end effector of the serial robot 106 is equipped with a target ball or other calibration device for pose measurement by a target sensor, used to cooperate with the target sensor to acquire the measured pose of the end effector in the measurement coordinate system during the measurement process.

[0030] The serial robot 106 is equipped with a motion controller, either internally or externally. The motion controller is used to receive corrected geometric parameters or an updated positive kinematics model to determine the end-effector pose using the updated positive kinematics model.

[0031] It should be noted that the end-effector pose calibration method for the serial robot provided in this application embodiment can generally be executed by server 105. Correspondingly, the end-effector pose calibration system for the serial robot provided in this application embodiment can generally be located in server 105. The end-effector pose calibration method for the serial robot provided in this application embodiment can also be executed by a server or server cluster that is different from server 105 and capable of communicating with the first terminal device 101, the second terminal device 102, the third terminal device 103, the serial robot 106, and / or server 105. Correspondingly, the end-effector pose calibration system for the serial robot provided in this application embodiment can also be located in a server or server cluster that is different from server 105 and capable of communicating with the first terminal device 101, the second terminal device 102, the third terminal device 103, the serial robot 106, and / or server 105.

[0032] It should be understood that Figure 1 The number of first terminal devices, second terminal devices, third terminal devices, serial robots, networks, and servers shown in the diagram is merely illustrative. Depending on implementation needs, any number of first terminal devices, second terminal devices, third terminal devices, serial robots, networks, and servers can be included.

[0033] The following will be based on Figure 1 The described scene, through Figures 2-3 The end-effector pose calibration method of the serial robot in the embodiments of the application is described in detail.

[0034] Figure 2 A flowchart of an end-effector pose calibration method for a serial robot according to an embodiment of this application is shown.

[0035] like Figure 2 As shown, the method includes operations S210 to S260.

[0036] In operation S210, in response to the triggered calibration task, the predicted pose of the end effector of the serial robot in the measurement coordinate system is determined by using the initial geometric parameters between each adjacent joint of the serial robot, the forward kinematic model, and the transformation relationship between the measurement coordinate system and the base coordinate system. The initial geometric parameters are used to describe the spatial pose relationship between adjacent joints.

[0037] In operation S220, the predicted pose is subtracted from the measured pose of the end in the measurement coordinate system measured by the target sensor to determine the target residual.

[0038] In operation S230, based on the error mapping relationship between the target residual and the initial geometric parameters on the pose of the end effector, the parameter correction amount of the initial geometric parameters between each adjacent joint is determined. The error mapping relationship is used to characterize the influence of the change of the initial geometric parameters on the pose of the end effector.

[0039] In operation S240, the initial geometric parameters are corrected based on the parameter correction amount to obtain the corrected geometric parameters.

[0040] In operation S250, in response to the fact that the corrected geometry parameters do not meet the stopping condition, the corrected geometry parameters are used as the initial geometry parameters for the next round, and the above steps are repeated.

[0041] In operation S260, in response to the correction of geometric parameters satisfying the stopping condition, the positive kinematics model is updated using the correction of geometric parameters so that the serial robot can use the updated positive kinematics model to determine the end-effector pose.

[0042] In response to a triggered calibration task, the current measured joint angles of each joint of the serial robot and the initial geometric parameters pre-stored in the robot controller are acquired. The initial geometric parameters may include link torsion angles, link lengths, joint offsets, and joint rotation angles between adjacent joints, which are used to describe the spatial pose relationship between the coordinate systems of adjacent joints.

[0043] In the embodiments of this application, the timing of triggering the calibration task is not specifically limited and can be flexibly set according to actual application needs. Optionally, the calibration task can be triggered before the serial robot leaves the factory or before it is put into use for the first time to complete the initial calibration of the robot's geometric parameters. In this case, the initial geometric parameters of the first iteration are determined according to the theoretical values ​​specified by the manufacturer.

[0044] Optionally, calibration tasks can also be triggered according to a preset time period. For example, a calibration task can be automatically triggered every preset running time (such as every 2 hours, every 8 hours, or every 24 hours) of the serial robot. In this case, the initial geometric parameters for the first iteration are the corrected geometric parameters that meet the stopping conditions determined in the previous calibration task.

[0045] Subsequently, the current joint angles and initial geometric parameters can be input or updated into the forward kinematics model. The forward kinematics model can calculate the predicted pose of the end effector in the base coordinate system according to the motion transmission law from the base to the end effector. Then, based on the pre-calibrated transformation relationship between the measurement coordinate system and the base coordinate system, the predicted pose of the end effector in the base coordinate system is transformed to the measurement coordinate system, resulting in the predicted pose of the end effector in the measurement coordinate system.

[0046] The actual pose of the end effector in the measurement coordinate system can be measured using target sensors such as trackers. The difference between the predicted pose and the actual pose is then used to determine the target residual. The target residual is used to quantify the deviation between the calculated and measured values ​​of the forward kinematics model under the current geometric parameters.

[0047] Based on the error mapping relationship between the target residual and the initial geometric parameters to the end-effector pose, the parameter correction amount of the initial geometric parameters between adjacent joints can be determined. The error mapping relationship is used to characterize the change in end-effector pose caused by a unit change in each geometric parameter, providing a mapping basis for inferring the geometric parameter correction amount from the residual.

[0048] Based on the parameter correction amount, the initial geometric parameters are corrected, and the parameter correction amount is superimposed on the initial geometric parameters of the current wheel to obtain the corrected geometric parameters.

[0049] If the corrected geometric parameters do not meet the stopping condition, the corrected geometric parameters can be used as the initial geometric parameters for the next round, and the above steps of determining the predicted pose, determining the target residual, determining the parameter correction amount, and correcting the geometric parameters can be repeated.

[0050] In response to the correction of geometric parameters satisfying the stopping condition, the correction of geometric parameters is used as the target geometric parameters and updated in the forward kinematics model so that the serial robot can use the updated forward kinematics model to determine the end-effector pose.

[0051] According to embodiments of this application, the stopping condition may include at least one of the following: the parameter correction amount of the current round is less than a first preset threshold; or, the target residual of the current round is less than a second preset threshold; or, the number of iterations of the current round reaches a preset maximum number of iterations; or the parameter correction amount is less than the first preset threshold and hardly changes anymore.

[0052] According to embodiments of this application, by introducing a transformation relationship between the measurement coordinate system and the base coordinate system, the predicted end-effector pose output by the forward kinematics model is transformed into the measurement coordinate system and subtracted from the actual end-effector pose measured by the target sensor. This achieves the measurement of the absolute end-effector pose and the calculation of residuals, improving the completeness of parameter identification information. Based on the error mapping relationship between the target residual and the initial geometric parameters to the end-effector pose, the correction amount of each geometric parameter is determined, and the initial geometric parameters are iteratively corrected until the stopping condition is met. This achieves systematic iterative calibration of the geometric parameters of the serial robot, enabling the corrected geometric parameters to accurately reflect the robot's true physical structure. Therefore, this at least partially solves the technical problems of high cost or poor positioning accuracy, achieving the technical effect of improving the absolute positioning accuracy of the robot.

[0053] According to an embodiment of this application, the predicted pose of the end effector of a serial robot in the measurement coordinate system is determined using the initial geometric parameters between adjacent joints, the forward kinematics model, and the transformation relationship between the measurement coordinate system and the base coordinate system. This includes: updating the current joint angles of each joint and the initial geometric parameters between adjacent joints in the forward kinematics model to determine the homogeneous transformation matrix between each adjacent joint coordinate system, wherein the homogeneous transformation matrix is ​​used to map the coordinates in the first joint coordinate system of the first joint to the second joint coordinate system of the second joint; multiplying the homogeneous transformation matrices between each adjacent joint coordinate system sequentially according to the motion transmission order from the base to the end effector of the serial robot to obtain the initial predicted pose of the end effector in the base coordinate system, where the base coordinate system is a coordinate system set on the base of the serial robot; and mapping the initial predicted pose from the base coordinate system to the measurement coordinate system according to the transformation relationship between the measurement coordinate system and the base coordinate system to obtain the predicted pose.

[0054] Figure 3 The spatial pose relationship between adjacent joints according to an embodiment of this application is shown.

[0055] like Figure 3 As shown, Figure 3 The diagram shows three adjacent joint axes of a serial robot, which are axes [axis name missing]. ,axis and shaft Among them, the shaft and shaft Let the two adjacent joint axes be axis i and axis i. The figures show the axes of two adjacent joints. The figure also shows the Z-axis of each joint axis, which corresponds to a coordinate system. , and .in, For the first The Z-axis of the joint coordinate system. Let Z be the coordinate axis of the i-th joint. For the first The Z-axis of the joint coordinate system. The direction of each joint axis is the Z-axis direction of the corresponding coordinate system.

[0056] Figure 3 The diagram also shows the direction of the common perpendicular between adjacent joint coordinate systems, i.e., the X-axis direction. For the first The direction of the common perpendicular between the joint coordinate system and the i-th joint coordinate system. Let the coordinate system of the i-th joint be the coordinate system of the i-th joint. The direction of the common perpendicular between the joint coordinate systems.

[0057] Figure 3The diagram also shows the link lengths between adjacent joint coordinate systems. For the first The length of the link between joint i and joint i, i.e., along The distance between the common perpendiculars of the directions; a i For the i-th joint and the i-th joint The length of the link between joints, i.e., along The distance between the common perpendiculars in the direction.

[0058] Figure 3 The joint offset d is also shown in the figure. i , indicating that the i-th joint along its Z-axis i Offset distance in the axial direction. And the joint rotation angle θ is shown. i , where represents the relative rotation angle of the i-th joint about its Z-axis direction.

[0059] During implementation, the current joint angles of each joint of the serial robot can be obtained. Then, the current joint angles and the initial geometric parameters between adjacent joints are input into the forward kinematics model. The forward kinematics model can be determined using a modified Denavit-Hartenberg (MD-H) model.

[0060] For example, let the direction of the robot's translation or rotation axis be the Z-axis, and the common perpendicular line between the axes be the X-axis. These are the link length, link torsion angle, joint rotation angle, and joint offset, respectively. The relationship between robot axes is decomposed into a superposition of translational and rotational transformations about the X-axis and about the Z-axis. First, the transformation about the X-axis... i-1 Rotation Angle, making Z i-1 and Z i Parallel, along X i-1 Translation Distance is used to make the origins of coordinate system {i-1} and coordinate system {i} coincide; then, the distance is used to make the origins of both .... i Rotation Angle, making X i-1 and X i Parallel, along Z i Translation Distance, such that X i-1 and X i Collinear. The homogeneous transformation matrix between the axes is:

[0061] (1);

[0062] in, Let represent the homogeneous transformation matrix from the (i-1)th axis to the (i)th axis. Indicates revolving around X i-1 Rotation , Indicates along X i-1 Translation , Indicates revolving around Z i Rotation , Indicates along Z i Translation , Indicates around along X i-1 Translation And rotate , Indicates around along Z i Translation And rotate .

[0063] Then, following the motion transmission sequence from the base to the end effector of the serial robot, the homogeneous transformation matrices between adjacent joint coordinate systems are multiplied sequentially to obtain the initial predicted pose of the end effector in the base coordinate system. The base coordinate system is a coordinate system fixed on the base of the serial robot.

[0064] Finally, based on the transformation relationship between the measurement coordinate system and the base coordinate system, the initial predicted pose is mapped from the base coordinate system to the measurement coordinate system to obtain the predicted pose.

[0065] {i} represents the coordinate system of the i-th joint of the robot; {B} represents the base coordinate system, which is the coordinate system defined based on the robot's base and is fixed. It generally refers to the initial coordinate system {0} of an n-axis robot; {W} is the wrist coordinate system, which is the coordinate system at the robot's wrist and generally refers to the last coordinate system {n} of an n-axis robot; {T} is the tool coordinate system or end effector coordinate system, which represents the coordinate system of the tool attached to the robot's end effector, such as a gripper or target ball; {M} is the measurement coordinate system, which represents the coordinate system of a measuring instrument, such as a laser tracker. All measurement data is based on the measurement coordinate system.

[0066] For a point in the terminal coordinate system {T} Its coordinates in the measurement coordinate system {M} are:

[0067] ;

[0068] in, ;

[0069] in, This indicates the predicted pose of the end effector in the measurement coordinate system. Identify the homogeneous transformation matrix from the end coordinate system to the measurement coordinate system. This represents the homogeneous transformation matrix from the base coordinate system to the measurement coordinate system. This represents the homogeneous transformation matrix from the wrist coordinate system to the base coordinate system (i.e., the calculation result of the forward kinematics model). Let represent the homogeneous transformation matrix from the tool coordinate system to the wrist coordinate system. Let represent the homogeneous transformation matrix from coordinate system {i} to coordinate system {i-1}, where i = 0, 1, 2…n.

[0070] According to the embodiments of this application, through the above coordinate system transformation steps, the position coordinates of the end in the end coordinate system are transferred to the measurement coordinate system step by step, realizing the coordinate system alignment between the output value of the positive kinematics model and the measured value of the target sensor, so that the predicted pose and the measured pose can be compared in the same coordinate system, providing more reliable error data for subsequent parameter identification.

[0071] According to an embodiment of this application, the error mapping relationship between the initial geometric parameters and the end-effector pose is obtained through the following steps: taking the partial derivatives of the initial geometric parameters in the homogeneous transformation matrix between adjacent joints of the serial robot to obtain the differential change of the homogeneous transformation matrix of each adjacent joint; constructing a local mapping matrix between each adjacent joint based on the differential change of each adjacent joint, wherein the local mapping matrix is ​​used to characterize the change in end-effector pose in the joint coordinate system caused by the unit change of the initial geometric parameters between adjacent joints; transforming the local mapping matrix between each adjacent joint from the joint coordinate system to the end-effector coordinate system to obtain the first contribution matrix of each adjacent joint in the end-effector coordinate system; transforming the first contribution matrix of each joint from the end-effector coordinate system to the measurement coordinate system to obtain the second contribution matrix of each adjacent joint in the measurement coordinate system; and concatenating the second contribution matrices of each adjacent joint in the measurement coordinate system in the joint order to obtain the error mapping relationship between the initial geometric parameters and the end-effector pose.

[0072] The error mapping relationship between initial geometric parameters and end-effector pose can be characterized by the Jacobian matrix. In implementation, the initial geometric parameters of the robot to be identified are typically... , The initial geometric parameters characterize the relationship between the (i-1)th joint and the ith joint. Therefore, the Jacobian matrix is ​​established using differential kinematics. It can be described as:

[0073] ;

[0074] in, Describe a point in the terminal coordinate system {T} The coordinates in the measurement coordinate system {M}.

[0075] The differential transformation from the base coordinate system to the tool coordinate system is:

[0076] ;

[0077] in, Let represent the differential change of the homogeneous transformation matrix of the i-th joint.

[0078] Therefore, for a small transformation of the end-effector pose, a small transformation can be obtained for each axis parameter:

[0079] ;

[0080] in, , For abbreviation, they correspond to respectively and .

[0081] Decomposition yields:

[0082] ;

[0083] in, This represents the minute displacement of the robot's end effector. This represents the minute amount of deflection at the robot's end effector; Let represent the local Jacobian matrix of the i-th joint, i.e., the local mapping matrix.

[0084] The differential variable of the i-th axis in coordinate system {i} All were transformed to the tool coordinate system to obtain , The adjoint matrix represents the coordinate system that transforms the differential variables from the tool coordinate system {T} to the joint axis coordinate system {i}. Therefore, we can conclude that:

[0085] ;

[0086] in, The matrix represents the influence of changes in the geometric parameters of the i-th joint on the position of the end effector in the measurement coordinate system. Represents the selection matrix. Represents a three-dimensional identity matrix. This is used to transform the local Jacobian matrix from the coordinate system of the i-th joint to the coordinate system of the end effector, so as to obtain the first contribution matrix of each joint in the tool coordinate system.

[0087] Therefore, the Jacobian matrix is:

[0088] ;

[0089] Where J is the complete Jacobian matrix, which is also the error mapping relationship between the initial geometric parameters and the end pose. Let represent the second contribution matrix of the i-th joint; the contribution matrices of each joint are horizontally concatenated in the order of the joints to obtain the complete Jacobian matrix.

[0090] According to embodiments of this application, a physical transfer model from the geometric parameter errors of each joint to the pose error of the end effector is constructed through the above steps. The local Jacobian matrix characterizes the quantitative relationship between the geometric parameter error of a single joint and the pose offset in the joint coordinate system. The adjoint matrix realizes the step-by-step transfer and accumulation of joint errors along the kinematic chain to the end effector. Finally, the complete Jacobian matrix obtained by stitching together establishes the overall mapping relationship between all joint geometric parameter errors and the position error of the end effector in the measurement coordinate system.

[0091] According to an embodiment of this application, there are m measured poses; each of the m measured poses has a corresponding predicted pose; there are m target residuals, where m is a positive integer greater than or equal to 1; wherein, based on the error mapping relationship between the target residuals and the initial geometric parameters to the end pose, the parameter correction amount of the initial geometric parameters between each adjacent joint is determined, including: stacking the m target residuals along the target direction to obtain a total residual vector; constructing a weighting matrix according to the measurement confidence corresponding to each measured pose and the dimensional difference between the pose component and attitude component in the measured pose, wherein the weighting matrix is ​​used to assign different weights to each target residual in the total residual vector; determining the weighted sum of squares of the total residual vector based on the weighting matrix, and constructing an objective function with the initial geometric parameters as optimization variables and minimizing the weighted sum of squares as the optimization objective; and determining the parameter correction amount based on the objective function and the error mapping relationship.

[0092] The initial geometric parameters of multiple adjacent joints can be spliced ​​together to obtain During implementation, the joint angles in group j measurements can be recorded using devices such as target sensors. And record the measured pose of the end effector as During implementation, parameters can be used. The predicted pose is obtained from the normal kinematic model and denoted as Therefore, the residual of a single measurement is:

[0093] ;

[0094] in, Let represent the residual of the j-th objective.

[0095] Stacking all m sets of target residuals yields the overall residual vector:

[0096] ;

[0097] Considering the differences in measurement confidence and dimensions for different measurement poses, a weighting matrix W can be introduced. This leads to an objective function that uses the initial geometric parameters as optimization variables and minimizes the weighted sum of squares as the optimization objective:

[0098] ;

[0099] Furthermore, by utilizing the error mapping relationship and the aforementioned objective function, the parameter correction amount can be determined.

[0100] According to the embodiments of this application, by stacking the target residuals of multiple sets of measurement data into a total residual vector, and introducing a weighting matrix to assign different weights to the position component and the attitude component respectively, the information of multiple measurement points is utilized to better balance the dimensional differences between the position residual and the attitude residual, making the subsequent iterative solution more accurate.

[0101] According to an embodiment of this application, determining the parameter correction amount based on the objective function and the error mapping relationship includes: determining the mapping data corresponding to the error mapping relationship between the current wheel and each adjacent joint of the current wheel based on the initial geometric parameters between them; performing a local linearization approximation on the objective function based on the mapping data to obtain a local approximation objective function for the current wheel; regularizing the local approximation objective function based on the damping regularization term and the prior term to obtain a regularized objective function for the current wheel, wherein the damping regularization term is determined according to a preset damping parameter, and the prior term is determined according to the difference between the initial geometric parameters of the current wheel and the first wheel; and solving the regularized objective function to obtain the parameter correction amount for the current wheel.

[0102] Due to the target residual The parameters exhibit a significantly nonlinear relationship; therefore, a nonlinear iterative algorithm must be used to solve the problem. For example, the iteration method is as follows:

[0103] ;

[0104] In the k-th iteration, the initial geometric parameters between each adjacent joint in the k-th round are first determined. First-order linearization is applied to the target residual:

[0105] ;

[0106] in, Indicates the parameter correction amount; This represents the mapping data for the k-th round. From this, we obtain the local approximate objective function corresponding to the k-th iteration step, or the k-th round:

[0107] ;

[0108] To suppress model linearization error and improve numerical stability, damping regularization terms and prior terms can be introduced into the parameter increments, which can be equivalently rewritten as:

[0109] ;

[0110] in, Refers to the initial geometric parameters of the first round. This represents the preset damping parameter for the k-th round. This represents the preset prior parameters for the k-th round.

[0111] According to the embodiments of this application, the target residual is linearized to the first order in each iteration, and a damping regularization term and a priori term are introduced to regularize the local approximate objective function, thereby suppressing numerical divergence when the matrix is ​​singular and improving the numerical stability of the iteration process.

[0112] According to an embodiment of this application, solving the regularization objective function to obtain the parameter correction amount for the current round includes: taking partial derivatives of the optimization variables in the regularization objective function to construct a system of equations corresponding to the regularization objective function; and solving the system of equations to obtain the parameter correction amount for the current round.

[0113] Solving for the extreme points using derivatives yields the corresponding linear equation:

[0114] ;

[0115] By solving the system of equations formed by the above equations, the parameter correction amount for the kth round can be obtained.

[0116] According to the embodiments of this application, by taking the partial derivative of the regularization objective function with respect to the optimization variables and setting the derivative to zero, the nonlinear optimization problem is transformed into a system of linear equations for solution, which simplifies the calculation process, improves the solution efficiency, and makes the calculation of parameter correction in each iteration fast and accurate.

[0117] According to embodiments of this application, robot calibration is achieved by compensating the robot's geometric parameters with corrected geometric parameters that meet the stopping conditions.

[0118] According to an embodiment of this application, the initial geometric parameters are corrected based on the parameter correction amount to obtain the corrected geometric parameters, including: adding the initial bonding parameters to the parameter correction amount to obtain the corrected geometric parameters.

[0119] According to the embodiments of this application, by superimposing the parameter correction amount onto the current wheel geometry parameter, the geometry parameter is updated wheel by wheel, so that the corrected geometry parameter gradually approaches the true physical value.

[0120] According to embodiments of this application, the initial geometric parameters include the following types of sub-parameters: link torsion angle, link length, joint offset, and joint rotation angle; wherein, link torsion angle represents the spatial angle between the axes of adjacent joints in the direction perpendicular to their common perpendicular; link length represents the spatial distance between the axes of adjacent joints in the direction of their common perpendicular; joint offset represents the offset distance of adjacent links in the direction of the joint axis; and joint rotation angle represents the relative rotation angle of adjacent links about the direction of the joint axis.

[0121] Figure 4 A structural block diagram of a serial robot according to an embodiment of this application is shown.

[0122] like Figure 4 As shown, the serial robot 106 includes: an end-effector pose calibration device 1061; wherein, the end-effector pose calibration device 1061 is used to perform the above-described end-effector pose calibration method for the serial robot.

[0123] Based on the aforementioned end-effector pose calibration method for serial robots, this application also provides an end-effector pose calibration system for serial robots. The following will combine... Figure 5 The device is described in detail.

[0124] Figure 5 A structural block diagram of an end-effector pose calibration system for a serial robot according to an embodiment of this application is shown.

[0125] like Figure 5 As shown, the end-effector pose calibration system 500 of the serial robot in this embodiment includes a first determining module 510, a calculation module 520, a second determining module 530, a correction module 540, an execution determining module 550, and an update module 560.

[0126] The first determining module 510 is used to determine the predicted pose of the end effector of the serial robot in the measurement coordinate system in response to the triggered calibration task, using the initial geometric parameters between each adjacent joint of the serial robot, the forward kinematic model, and the transformation relationship between the measurement coordinate system and the base coordinate system. The initial geometric parameters are used to describe the spatial pose relationship between adjacent joints. In one embodiment, the first determining module 510 can be used to perform the operation S210 described above, which will not be repeated here.

[0127] The calculation module 520 is used to calculate the difference between the predicted pose and the measured pose of the end in the measurement coordinate system measured by the target sensor to determine the target residual; in one embodiment, the first determination module 510 can be used to perform the operation S220 described above, which will not be repeated here.

[0128] The second determining module 530 is used to determine the parameter correction amount of the initial geometric parameters between each adjacent joint based on the error mapping relationship between the target residual and the initial geometric parameters to the pose of the end effector. The error mapping relationship is used to characterize the influence of the change of the initial geometric parameters on the pose of the end effector. In one embodiment, the first determining module 510 can be used to perform the operation S230 described above, which will not be repeated here.

[0129] The correction module 540 is used to correct the initial geometric parameters based on the parameter correction amount to obtain the corrected geometric parameters; in one embodiment, the first determination module 510 can be used to perform the operation S240 described above, which will not be repeated here.

[0130] The execution determination module 550 is used to, in response to the fact that the corrected geometric parameters do not meet the stopping condition, use the corrected geometric parameters as the initial geometric parameters for the next round and repeat the above steps; in one embodiment, the first determination module 510 can be used to execute the operation S250 described above, which will not be repeated here.

[0131] The update module 560 is used to update the forward kinematics model using the corrected geometric parameters in response to the stopping condition being met, so that the serial robot can use the updated forward kinematics model to determine the end-effector pose. In one embodiment, the first determination module 510 can be used to perform the operation S260 described above, which will not be repeated here.

[0132] According to embodiments of this application, any multiple modules among the first determining module 510, calculation module 520, second determining module 530, correction module 540, execution determining module 550, and update module 560 can be combined into one module, or any one of these modules can be split into multiple modules. Alternatively, at least some of the functions of one or more of these modules can be combined with at least some of the functions of other modules and implemented in one module. According to embodiments of this application, at least one of the first determining module 510, calculation module 520, second determining module 530, correction module 540, execution determining module 550, and update module 560 can be at least partially implemented as hardware circuitry, such as a field-programmable gate array (FPGA), a programmable logic array (PLA), a system-on-a-chip, a system-on-a-substrate, a system-on-package, an application-specific integrated circuit (ASIC), or implemented in hardware or firmware by any other reasonable means of integrating or packaging the circuitry, or implemented in any one of the three implementation methods of software, hardware, and firmware, or in a suitable combination of any of these. Alternatively, at least one of the first determining module 510, the calculation module 520, the second determining module 530, the correction module 540, the execution determining module 550, and the update module 560 may be implemented at least partially as a computer program module, which can perform corresponding functions when the computer program module is run.

[0133] Figure 6 A block diagram of an electronic device suitable for implementing an end-effector pose calibration method for a serial robot, according to an embodiment of this application, is shown.

[0134] like Figure 6 As shown, an electronic device 600 according to an embodiment of this application includes a processor 601, which can perform various appropriate actions and processes according to a program stored in a read-only memory ROM 602 or a program loaded from a storage portion 608 into a random access memory RAM 603. The processor 601 may include, for example, a general-purpose microprocessor (e.g., a CPU), an instruction set processor and / or an associated chipset and / or a special-purpose microprocessor (e.g., an application-specific integrated circuit (ASIC)), etc. The processor 601 may also include onboard memory for caching purposes. The processor 601 may include a single processing unit or multiple processing units for performing different actions of the method flow according to an embodiment of this application.

[0135] RAM 603 stores various programs and data required for the operation of electronic device 600. Processor 601, ROM 602, and RAM 603 are interconnected via bus 604. Processor 601 executes various operations of the method flow according to embodiments of this application by executing programs in ROM 602 and / or RAM 603. It should be noted that the programs may also be stored in one or more memories other than ROM 602 and RAM 603. Processor 601 may also execute various operations of the method flow according to embodiments of this application by executing programs stored in said one or more memories.

[0136] According to embodiments of this application, the electronic device 600 may further include an input / output (I / O) interface 605, which is also connected to a bus 604. The electronic device 600 may also include one or more of the following components connected to the input / output (I / O) interface 605: an input section 606 including a keyboard, mouse, etc.; an output section 607 including a cathode ray tube (CRT), liquid crystal display (LCD), etc., and a speaker, etc.; a storage section 608 including a hard disk, etc.; and a communication section 609 including a network interface card such as a LAN card, modem, etc. The communication section 609 performs communication processing via a network such as the Internet. A drive 610 is also connected to the input / output (I / O) interface 605 as needed. A removable medium 611, such as a disk, optical disk, magneto-optical disk, semiconductor memory, etc., is installed on the drive 610 as needed so that computer programs read from it can be installed into the storage section 608 as needed.

[0137] This application also provides a computer-readable storage medium, which may be included in the device / apparatus / system described in the above embodiments; or it may exist independently and not assembled into the device / apparatus / system. The computer-readable storage medium carries one or more programs, which, when executed, implement the method according to the embodiments of this application.

[0138] According to embodiments of this application, the computer-readable storage medium can be a non-volatile computer-readable storage medium, such as including but not limited to: portable computer disks, hard disks, random access memory (RAM), read-only memory (ROM), erasable programmable read-only memory (EPROM or flash memory), portable compact disk read-only memory (CD-ROM), optical storage devices, magnetic storage devices, or any suitable combination thereof. In this application, the computer-readable storage medium can be any tangible medium containing or storing a program that can be used by or in conjunction with an instruction execution system, apparatus, or device. For example, according to embodiments of this application, the computer-readable storage medium may include ROM 602 and / or RAM 603 and / or one or more memories other than ROM 602 and RAM 603 described above.

[0139] Embodiments of this application also include a computer program product comprising a computer program containing program code for performing the methods shown in the flowchart. When the computer program product is run on a computer system, the program code is used to cause the computer system to implement the methods provided in the embodiments of this application.

[0140] When the computer program is executed by the processor 601, it performs the functions defined in the system / apparatus of this application embodiment. According to the embodiments of this application, the systems, apparatuses, modules, units, etc., described above can be implemented by computer program modules.

[0141] In one embodiment, the computer program may rely on a tangible storage medium such as an optical storage device or a magnetic storage device. In another embodiment, the computer program may also be transmitted and distributed in the form of signals over a network medium, and downloaded and installed via the communication section 609, and / or installed from the removable medium 611. The program code contained in the computer program can be transmitted using any suitable network medium, including but not limited to: wireless, wired, etc., or any suitable combination thereof.

[0142] In such an embodiment, the computer program can be downloaded and installed from a network via the communication section 609, and / or installed from the removable medium 611. When the computer program is executed by the processor 601, it performs the functions defined in the system of this application embodiment. According to the embodiments of this application, the systems, devices, apparatuses, modules, units, etc., described above can be implemented by computer program modules.

[0143] According to embodiments of this application, program code for executing the computer programs provided in the embodiments of this application can be written in any combination of one or more programming languages. Specifically, these computational programs can be implemented using high-level procedural and / or object-oriented programming languages, and / or assembly / machine languages. Programming languages ​​include, but are not limited to, languages ​​such as Java, C++, Python, "C", or similar programming languages. The program code can be executed entirely on the user's computing device, partially on the user's device, partially on a remote computing device, or entirely on a remote computing device or server. In cases involving remote computing devices, the remote computing device can be connected to the user's computing device via any type of network, including a local area network (LAN) or a wide area network (WAN), or it can be connected to an external computing device (e.g., via the Internet using an Internet service provider).

[0144] The flowcharts and block diagrams in the accompanying drawings illustrate the architecture, functionality, and operation of possible implementations of systems, methods, and computer program products according to various embodiments of this application. In this regard, each block in a flowchart or block diagram may represent a module, segment, or portion of code containing one or more executable instructions for implementing a specified logical function. It should also be noted that in some alternative implementations, the functions indicated in the blocks may occur in a different order than those indicated in the drawings. For example, two consecutively indicated blocks may actually be executed substantially in parallel, and they may sometimes be executed in reverse order, depending on the functions involved. It should also be noted that each block in a block diagram or flowchart, and combinations of blocks in a block diagram or flowchart, may be implemented using a dedicated hardware-based system that performs the specified function or operation, or using a combination of dedicated hardware and computer instructions.

[0145] Those skilled in the art will understand that the features described in the various embodiments of this application can be combined and / or combined in various ways, even if such combinations or combinations are not explicitly described in this application. In particular, the features described in the various embodiments of this application can be combined and / or combined in various ways without departing from the spirit and teachings of this application. All such combinations and / or combinations fall within the scope of this application.

[0146] The embodiments of this application have been described above. However, these embodiments are merely illustrative and not intended to limit the scope of this application. Although various embodiments have been described above, this does not mean that the measures in the various embodiments cannot be used advantageously in combination. Without departing from the scope of this application, those skilled in the art can make various substitutions and modifications, all of which should fall within the scope of this application.

Claims

1. A method for end-effector pose calibration of a serial robot, characterized in that, The method includes: In response to the triggered calibration task, the predicted pose of the end effector of the serial robot in the measurement coordinate system is determined by using the initial geometric parameters between each adjacent joint of the serial robot, the positive kinematic model, and the transformation relationship between the measurement coordinate system and the base coordinate system. The initial geometric parameters are used to describe the spatial pose relationship between adjacent joints. The target residual is determined by subtracting the predicted pose from the measured pose of the end in the measurement coordinate system as measured by the target sensor. Based on the error mapping relationship between the target residual and the initial geometric parameters on the pose of the end effector, the parameter correction amount of the initial geometric parameters between each adjacent joint is determined. The error mapping relationship is used to characterize the influence of the change of the initial geometric parameters on the pose of the end effector. Based on the parameter correction amount, the initial geometric parameters are corrected to obtain the corrected geometric parameters; In response to the fact that the corrected geometric parameters do not meet the stopping condition, the corrected geometric parameters are used as the initial geometric parameters for the next round, and the above steps are repeated. In response to the correction geometric parameters satisfying the stopping condition, the positive kinematics model is updated using the correction geometric parameters so that the serial robot can use the updated positive kinematics model to determine the end-effector pose.

2. The end-effector pose calibration method according to claim 1, characterized in that, The method of determining the predicted pose of the end effector of the serial robot in the measurement coordinate system by utilizing the initial geometric parameters between adjacent joints of the serial robot, the forward kinematics model, and the transformation relationship between the measurement coordinate system and the base coordinate system includes: The current joint angles of each joint of the serial robot and the initial geometric parameters between each adjacent joint are updated in the forward kinematics model to determine the homogeneous transformation matrix between each adjacent joint coordinate system. The homogeneous transformation matrix is ​​used to map the coordinates in the first joint coordinate system of the first joint in the adjacent joint to the second joint coordinate system of the second joint. According to the motion transmission sequence from the base of the serial robot to the end effector, the homogeneous transformation matrices between each adjacent joint coordinate system are multiplied in turn to obtain the initial predicted pose of the end effector in the base coordinate system, where the base coordinate system is a coordinate system set on the base of the serial robot. Based on the transformation relationship between the measurement coordinate system and the base coordinate system, the initial predicted pose is mapped from the base coordinate system to the measurement coordinate system to obtain the predicted pose.

3. The end-effector pose calibration method according to claim 1, characterized in that, The measured poses are m; each of the m measured poses has a corresponding predicted pose; the target residuals are m, where m is a positive integer greater than or equal to 1; The step of determining the parameter correction amount of the initial geometric parameters between adjacent joints based on the error mapping relationship between the target residual and the initial geometric parameters to the end-effector pose includes: Stack the m target residuals along the target direction to obtain the overall residual vector; Based on the measurement confidence corresponding to each of the measurement poses and the dimensional differences between the pose components and attitude components in the measurement poses, a weighting matrix is ​​constructed, wherein the weighting matrix is ​​used to assign different weights to each target residual in the overall residual vector; The weighted sum of squares of the total residual vector is determined based on the weighted matrix, and an objective function is constructed with the initial geometric parameters as optimization variables and the minimization of the weighted sum of squares as the optimization objective. The parameter correction amount is determined based on the objective function and the error mapping relationship.

4. The end-effector pose calibration method according to claim 3, characterized in that, Determining the parameter correction amount based on the objective function and the error mapping relationship includes: Based on the initial geometric parameters between each adjacent joint of the current wheel, determine the mapping data corresponding to the error mapping relationship between the current wheel; Based on the mapping data, the objective function is approximated by local linearization to obtain the local approximate objective function for the current round; Based on the damping regularization term and the prior term, the local approximate objective function is regularized to obtain the regularized objective function of the current wheel. The damping regularization term is determined according to the preset damping parameter, and the prior term is determined according to the difference between the initial geometric parameters of the current wheel and the first wheel. Solve the regularization objective function to obtain the parameter correction amount for the current round.

5. The end-effector pose calibration method according to claim 4, characterized in that, Solving the regularization objective function to obtain the parameter correction amount for the current round includes: Calculate the partial derivatives of the optimization variables in the regularization objective function to construct the system of equations corresponding to the regularization objective function; Solve the system of equations to obtain the parameter correction amount for the current round.

6. The end-effector pose calibration method according to claim 1, characterized in that, The step of correcting the initial geometric parameters based on the parameter correction amount to obtain the corrected geometric parameters includes: The initial bonding parameters are added to the parameter correction amount to obtain the corrected geometric parameters.

7. The end-effector pose calibration method according to claim 1, characterized in that, The stopping condition includes at least one of the following: The parameter correction amount in the current round is less than a first preset threshold; or, The target residual in the current round is less than a second preset threshold; or, The current iteration count has reached the preset maximum iteration count.

8. The end-effector pose calibration method according to claim 1, characterized in that, The error mapping relationship between the initial geometric parameters and the end-effector pose is obtained through the following steps: The partial derivatives of the initial geometric parameters in the homogeneous transformation matrix between each adjacent joint of the serial robot are obtained to obtain the differential changes of the homogeneous transformation matrix between each adjacent joint. Based on the differential change of each of the adjacent joints, a local mapping matrix is ​​constructed between each of the adjacent joints, wherein the local mapping matrix is ​​used to characterize the change in end pose in the joint coordinate system caused by the unit change of the initial geometric parameters between the adjacent joints. The local mapping matrix between each of the adjacent joints is transformed from the joint coordinate system of each joint to the end-effector coordinate system to obtain the first contribution matrix of each of the adjacent joints in the end-effector coordinate system. The first contribution matrix of each joint is transformed from the end coordinate system to the measurement coordinate system to obtain the second contribution matrix of each adjacent joint in the measurement coordinate system. The second contribution matrices of each adjacent joint in the measurement coordinate system are sequentially spliced ​​together according to the joint order to obtain the error mapping relationship between the initial geometric parameters and the end pose.

9. The end-effector pose calibration method according to claim 1, characterized in that, The initial geometric parameters include the following types of sub-parameters: link torsion angle, link length, joint offset, and joint rotation angle; Wherein, the link torsion angle represents the spatial angle between the axes of the adjacent joints in the direction perpendicular to their common perpendicular; the link length represents the spatial distance between the axes of the adjacent joints in the direction of their common perpendicular; the joint offset represents the offset distance of the adjacent links in the direction of the joint axis; and the joint rotation angle represents the relative rotation angle of the adjacent links about the direction of the joint axis.

10. A serial robot, characterized in that, include: End-effector pose calibration device; The end-effector pose calibration device is used to perform the end-effector pose calibration method of the serial robot as described in any one of claims 1 to 9.

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