A step-by-step error modeling calibration method for series-arms with closed-loop sub-chain

Abstract

The application relates to the technical field of robot control, in particular to a step-by-step error modeling and calibration method for a serial arm with a closed-loop subchain, which comprises the following steps: step-by-step establishment of a geometric error model of the closed-loop subchain and a terminal error mapping model of the serial robot, and obtaining a new error parameter calibration algorithm for the serial robot with the closed-loop subchain based on a laser tracker. The application measures the pose error existing in the working space of the mechanical arm caused by various factors such as structural error and measurement error of the mechanical arm by using a high-precision laser tracker device, maps the error to the error model of the serial arm with the closed-loop subchain, thereby obtaining the real structural parameters of the serial arm with the closed-loop subchain, compensating the kinematic model, and controlling the robot according to the new kinematic model, so that the motion error of the robot terminal pose is reduced.

Description

Technical Field

[0001] This invention relates to the field of robot control technology, specifically to a step-by-step error modeling and calibration method for serial arms with closed-loop subchains. Background Technology

[0002] Robotic machining, with its advantages of large workspace, high flexibility, and low cost, is considered a promising technology in modern manufacturing. However, the inherent low absolute positioning accuracy and low stiffness resulting from the serial structure of serial industrial robots limit their widespread application in industry. The latter often causes undesirable static and even dynamic deformation errors in robotic machining. To increase system stiffness, many robot manufacturers offer specialized industrial robots with closed-loop subchains for machining applications. While this improves the stiffness of such robots, their absolute positioning accuracy within the workspace is typically only a few millimeters, failing to meet the requirements of manufacturing applications. One approach to address this issue is to refine the theoretical kinematic model through kinematic calibration, thereby improving the robot's absolute positioning accuracy.

[0003] In recent years, with the explosive application of robotic arms in domestic automated production lines and the continuous advancement of the "machine replacement project," the importance of kinematic calibration and error compensation for robots has become increasingly prominent. Summary of the Invention

[0004] To address the shortcomings of existing technologies, the present invention aims to provide a step-by-step error modeling and calibration method for serial arms with closed-loop subchains.

[0005] To achieve the above objectives, the present invention provides the following technical solution: a step-by-step error modeling and calibration method for serial arms with closed-loop subchains, the steps of which are as follows:

[0006] (1) Based on the existing serial robot model with closed-loop subchains, a serial arm model without closed-loop subchains is formed by operation. The kinematics of the model are derived. Then, the error calibration model of the serial robot without subchains is established based on the differential motion equation. Then, closed-loop constraint equations are established for the two closed-loop subchains respectively. The error models of the structural parameter error of the closed-loop subchains and the rotation angle error of the passive joints on the main chain are derived. Finally, the error models of the two closed-loop subchains are added to the previously established error model of the serial robot without subchains to obtain the overall error parameter calibration model of the serial arm with closed-loop subchains.

[0007] (2) Determine a spiral trajectory within the robot's workspace, with a spiral radius of r and a pitch of h. Verify the feasibility of the trajectory using a robot simulation model, and then conduct a calibration experiment.

[0008] (3) Set the robot's motion trajectory in the robot control software according to the given robot motion trajectory, drive the robot to perform spiral motion, and verify that its actual motion trajectory is consistent with the original motion trajectory to verify the correctness of the model.

[0009] (4) In order to construct the base coordinate system of the robotic arm, three target ball seats are placed at specific positions on the base plane of the robotic arm. After measuring the position of the target ball on it with a laser tracker, the center point of the three points is taken as the origin of the coordinate system, the direction of the measurement point from the origin to the robotic arm is taken as the x-axis, the upward direction of the normal of the measurement point plane is taken as the z-axis, and the y-axis is determined according to the right-hand rule, thereby constructing a global base coordinate system.

[0010] (5) Install three target ball seats at the end of the robot, start the robot and execute the robot motion trajectory that has been set, so that the end effector moves along the spiral trajectory, read the joint drive value corresponding to each movement point when the end of the robot is making spiral motion and monitor the motion state of the end of the robot in real time, and use a laser tracker to record the spatial position of the three target balls at the end of the trajectory under the base target system.

[0011] (6) Read the saved experimental data in MATLAB software, process the data to obtain the actual spatial pose of the robot end effector, and then substitute the driving value corresponding to the trajectory into the forward kinematics equation of the robot to calculate the theoretical spatial pose of the robot end effector.

[0012] (7) Based on step (6), in MTATLAB, the difference between the actual spatial pose and the theoretical spatial pose is calculated according to the established error mapping model to obtain the end pose error. Then, the error is substituted into the error mapping model to identify the actual structural parameter error of the robot. Subsequently, the identified structural parameter error is added to the structural parameters in the theoretical model to compensate for the robot's motion error. The corrected model is then substituted into the experiment, and the above steps are repeated to verify the accuracy and effectiveness of the identified parameters.

[0013] In some embodiments, in step (2), the set spiral trajectory is based on the x-axis direction of the base coordinate system, and the spiral radius and pitch are determined according to the robot's workspace, thereby realizing the robot's movement in space.

[0014] In some embodiments, in step (5), the continuous motion of the robot in space is discretized, that is, it needs to be stationary and stable during the motion so that the laser tracker can measure the position when the robot is stable.

[0015] In some embodiments, the derivation of the main kinematic chain error mapping model is as follows:

[0016] The main motion chain of this mechanism is a series arm model that forms a series arm without closed-loop sub-chains. It is a three-degree-of-freedom series robotic arm. A fixed Cartesian coordinate system is established in the mechanism.

[0017] By performing forward kinematics derivation on the main kinematic chains, we obtain the joint drive values ​​in the fixed coordinate system O-XYZ and their transformation to the real-time pose of the center of the end-effector coordinate system. The relation, and its forward kinematic expression, are as follows:

[0018]

[0019] In the formula: s(), c(), t() represent the trigonometric functions sin(), cos(), and tan(), respectively. θ 12 It is an abbreviation for θ1+θ2;

[0020] The mapping relationship between the pose error and structural parameter error of the tandem arm is derived based on the differential method. This mapping relationship is obtained by taking the total differential of both sides of the forward kinematics equation, as follows:

[0021]

[0022] Simplifying the above equations, we obtain the mapping relationship between the end-effector pose error and the kinematic error parameters of the main chain:

[0023]

[0024] In the formula: δx, δy, δz, δβ, δγ are the end-effector pose errors.

[0025] δh, δL2, δL6, δd1, δd7, δθ0, δθ1, δθ2 are the structural errors of the main chain;

[0026] This establishes the mapping relationship between the end-effector pose error and the main chain structure parameter errors:

[0027] δpose=J0δr0

[0028] In the formula: δpose=[δx, δy, δz, δβ, δγ] T ,

[0029] δr=[δh, δL2, δL6, δd1, δd7, δθ0, δθ1, δθ2] T .

[0030] In some embodiments, the derivation of the closed-loop subchain error mapping model is as follows:

[0031] When there are errors in the link length and the driven joint variables, the actual passive joint variables are calculated for the closed-loop sub-chain structure, which includes the first closed-loop sub-chain structure and the second closed-loop sub-chain structure.

[0032] First, let's define the closed-loop constraint equation for the first closed-loop subchain as follows:

[0033]

[0034] The mapping relationship between the passive joint variable error and the closed-loop structural parameter error of the tandem arm is derived based on the differential method. This mapping relationship can be obtained by applying the closed-loop vector equation E 1x and E 1z The total differential of both sides is obtained as follows:

[0035]

[0036] Simplifying the above formulas, we obtain the mapping relationship between the error of the first passive joint variable in the main chain and the structural error of the first closed-loop sub-chain:

[0037]

[0038] In the formula: δθ1 is the error of the first passive joint variable of the main chain, and δL1,δL4,δd3,δd4,δq1 are the structural parameter errors of the first closed-loop sub-chain.

[0039] This yields the mapping relationship between the error of the first passive joint variable and the error of the first closed-loop subchain structure parameters:

[0040]

[0041] In the formula: δr1=[δL1, δL4, δd3, δd4, δq1] T ,

[0042] Similarly, let's define the closed-loop constraint equations for the second closed-loop subchain:

[0043]

[0044] The mapping relationship between the passive joint variable error and the closed-loop structural parameter error of the tandem arm is derived based on the differential method. This mapping relationship can be obtained by applying the closed-loop vector equation E 2x and E 2z The total differential of both sides is obtained as follows:

[0045]

[0046] Rearranging the above formulas into matrix form yields the mapping between the error of the second passive joint variable in the main chain and the structural error of the second closed-loop sub-chain:

[0047]

[0048] In the formula: δθ2 is the error of the second passive joint variable of the main chain, and δh1,δd5,δβ0,δd6,δL5,δd2,δq2 are the errors of the structural parameters of the second closed-loop sub-chain.

[0049] This yields the mapping relationship between the second passive joint variable error and the second closed-loop subchain structural parameter error:

[0050]

[0051] In the formula: δr2=[δh1, δd5, δβ0, δd6, δL5, δd2, δq2] T ,

[0052] Thus, the error mapping model for the two closed-loop subchains was obtained.

[0053] In some embodiments, the method for establishing and simulating the overall error mapping model of the robotic arm with closed-loop subchains is as follows:

[0054] By applying the mapping relationship between passive joint variable errors and structural parameter errors of closed-loop subchains to the error mapping model of the main kinematic chain of the tandem arm, the overall error mapping model can be obtained as follows:

[0055]

[0056] δpose=Jδr

[0057] In the formula: δr=[δh, δL2, δL6, δd1, δd7, δθ0, δr1, δr2] T .

[0058] Compared with the prior art, the beneficial effects of the present invention are:

[0059] 1. This invention proposes an error parameter mapping model for a robotic arm with a closed-loop subchain. This model effectively models the error parameters of the robotic arm based on a laser tracker and ensures the least-squares solution of the error parameters. This not only improves the calibration accuracy of the robot but also enhances its stability and repeatability in continuous or repetitive operations. Furthermore, the use of a least-squares iterative strategy with regularization terms further reduces the phenomenon of model overfitting, making the error parameters of the calibrated robot more accurate and significantly improving the reliability of the model.

[0060] 2. In this implementation case, the error parameter modeling of the serial arm with closed-loop subchain is mainly achieved by establishing a comprehensive error parameter mapping model and a kinematic error mapping model that combines the kinematic error mapping models of the serial manipulator and the closed-loop subchain. The accuracy of the identification model is optimized through iterative calculations, thereby improving the error parameter identification accuracy of the serial arm with closed-loop subchain. After kinematic parameter identification, it is ensured that the serial arm with closed-loop subchain can maintain a better high-precision standard during operation.

[0061] Details of one or more embodiments of this application are set forth in the following drawings and description to make other features, objects and advantages of this application more readily apparent. The embodiments of this application will provide a detailed description and understanding of the application. Attached Figure Description

[0062] Figure 1 Schematic diagram of a robotic arm with a closed-loop sub-chain;

[0063] Figure 2 This is a schematic diagram of the first closed-loop structure of a robotic arm with a closed-loop sub-chain.

[0064] Figure 3 This is a schematic diagram of the second closed-loop structure of a robotic arm with a closed-loop sub-chain.

[0065] Figure 4 This is the overall flowchart for calibrating the error parameters of the robotic arm with closed-loop subchain based on a laser tracker in this implementation case. Detailed Implementation

[0066] The technical solutions of the embodiments of the present invention will be clearly and completely described below with reference to the accompanying drawings. Obviously, the described embodiments are only some embodiments of the present invention, and not all embodiments. Based on the embodiments of the present invention, all other embodiments obtained by those skilled in the art without creative effort are within the scope of protection of the present invention.

[0067] Some existing studies have derived error models in closed-loop systems based on perfect parallelogram mechanisms. However, the errors of passive joint variables depend only linearly on the errors of driving joint variables and do not model the errors of structural parameters in the closed loop, thus failing to satisfy the completeness of the error mapping model. Therefore, it is necessary to establish an error mapping model that includes all structural errors in the closed loop.

[0068] This invention provides a technical solution: a step-by-step error modeling and calibration method for a serial arm with a closed-loop subchain. By using a high-precision laser tracker, the pose error of the robotic arm in the workspace caused by various factors such as its own structural errors and measurement errors is measured, and this error is mapped to the error model of the serial arm with the closed-loop subchain, thereby obtaining its true structural parameters to compensate the kinematic model. Then, the robot is controlled according to the new kinematic model, thereby reducing the motion error of the robot's end effector pose.

[0069] See Figure 4 As shown, it includes the following steps:

[0070] 1) Based on the existing serial robot model with closed-loop subchains, the closed-loop subchains can be virtually cut to form a serial arm model without closed-loop subchains. Kinematic derivation is then performed on this model, and an error calibration model for the serial robot without subchains is established based on the differential equations of motion. Next, closed-loop constraint equations are established for each of the two closed-loop subchains, deriving the error models for the structural parameter errors of the closed-loop subchains and the rotation angle errors of the passive joints on the main chain. Finally, the error models of the two closed-loop subchains are added to the previously established error model of the serial robot arm without subchains, thus obtaining the overall error parameter calibration model for the serial arm with closed-loop subchains.

[0071] 2) Determine a helical trajectory within the robot's workspace, with a helical radius of r and a pitch of h. Verify the feasibility of the trajectory using a robot simulation model, and then conduct a calibration experiment.

[0072] 3) Write a motion control program for the robot based on the given robot motion trajectory in the robot control software. Use the written program to drive the robot to perform helical motion and verify that its actual motion trajectory is consistent with the original motion trajectory, thus verifying the correctness of the model.

[0073] 4) To construct the base coordinate system of the robotic arm, three target ball mounts are placed at specific positions on the base plane of the robotic arm. After measuring the position of the target ball mounts with a laser tracker, the center point of the three points is taken as the origin of the coordinate system, the direction of the measurement point from the origin to the robotic arm is taken as the x-axis, the upward direction of the normal to the plane of the measurement point is taken as the z-axis, and the y-axis is determined according to the right-hand rule, thus constructing a global base coordinate system.

[0074] 5) Install three target ball seats at the end of the robot, start the robot and execute the pre-programmed program to make the end effector move along a spiral trajectory. Read the joint drive value corresponding to each movement point of the robot end when it is making spiral motion and monitor the motion state of the robot end in real time. Use a laser tracker to record the spatial position of the three target balls at the end of the trajectory under the base target system.

[0075] 6) Read the saved experimental data in MATLAB software, process the data, and obtain the actual spatial pose of the robot's end effector. Then, according to the robot's forward kinematics equations, substitute the driving values ​​corresponding to the trajectory to calculate the theoretical spatial pose of the robot's end effector.

[0076] 7) Based on step 6), in MTATLAB, the difference between the actual spatial pose and the theoretical spatial pose is calculated according to the established error mapping model to obtain the end-effector pose error. This error is then substituted into the error mapping model to identify the actual structural parameter error of the robot. Subsequently, the identified structural parameter error is added to the structural parameters in the theoretical model to compensate for the robot's motion error. The corrected model is then substituted into the experiment, and the above steps are repeated to verify the accuracy and effectiveness of the identified parameters.

[0077] In step 2), the set spiral trajectory is based on the x-axis of the base coordinate system. The spiral radius and pitch are determined according to the robot's workspace, thereby realizing the robot's movement in space.

[0078] In step 5), the continuous motion of the robot in space is discretized, that is, it needs to be stationary and stable during the motion so that the laser tracker can measure the position when the robot is stable.

[0079] The error modeling involved in this invention is characterized by the fact that the rotational joints of the main chain are passive joints, which are different from the structural errors caused by the complex structure itself in the active joints. It can be considered that the structural errors of the passive joints on the main chain are caused by the structural errors of the closed-loop sub-chains. Therefore, a mapping model from the end-effector pose error to the structural error of the main kinematic chain can be established first, and then a mapping model from the structural error of the closed-loop sub-chain to the structural error of the main chain joints can be established. Thus, a mapping model from the end-effector pose error to the overall structural error of the robot can be obtained.

[0080] The technical solution of the present invention will be further described below with reference to the accompanying drawings.

[0081] The present invention includes the following steps:

[0082] Derivation of the main kinematic chain error mapping model

[0083] The closed-loop sub-chain of the mechanism targeted by this invention is virtually cut apart to form a serial arm model without closed-loop sub-chains, which is the main motion chain of this mechanism. It is a three-degree-of-freedom serial robotic arm. A projection diagram with a fixed Cartesian coordinate system is established in the mechanism for reference. Figure 1 As shown, the X-axis is collinear with O1A2, the Z-axis is along the O0O1 direction, the Y-axis satisfies the right-hand rule, and the origin of the end-tool coordinate system Ot-XYZ is the midpoint of O3, where the end-tool coordinate system and the base coordinate system have the same direction.

[0084]

[0085] By performing forward kinematics derivation on the main kinematic chains, we obtain the joint drive values ​​in the fixed coordinate system O-XYZ and their transformation to the real-time pose of the center of the end-effector coordinate system. The relation, and its forward kinematic expression, are as follows:

[0086] In the formula: s(), c(), t() represent the trigonometric functions sin(), cos(), and tan(), respectively. θ 12 It is an abbreviation for θ1+θ2.

[0087] The mapping relationship between the pose error and structural parameter error of the tandem arm is derived based on the differential method. This mapping relationship can be obtained by taking the total differential of both sides of the forward kinematics equation, as shown below:

[0088]

[0089] Simplifying the above equations, we obtain the mapping relationship between the end-effector pose error and the kinematic error parameters of the main chain:

[0090] In the formula: δx, δy, δz, δβ, δγ are the end-effector pose errors.

[0091] δh, δL2, δL6, δd1, δd7, δθ0, δθ1, and δθ2 represent the structural errors of the main chain.

[0092] This establishes the mapping relationship between the end-effector pose error and the main chain structure parameter errors:

[0093] δpose=J0δr0

[0094] In the formula: δpose=[δx, δy, δz, δβ, δγ]T,

[0095] δr=[δh, δL2, δL6, δd1, δd7, δθ0, δθ1, δθ2]T.

[0096] (2) Derivation of the closed-loop subchain error mapping model.

[0097] When there are errors in the link length and the driven joint variables, the actual passive joint variables should be adjusted accordingly. Figure 2 , Figure 3 The calculation is performed on the closed-loop subchain structure shown.

[0098] like Figure 2 The simplified diagram of the first closed-loop subchain structure is shown below. The closed-loop constraint equation for the first closed-loop subchain is first defined as follows:

[0099]

[0100] The mapping relationship between the passive joint variable error and the closed-loop structural parameter error of the tandem arm is derived based on the differential method. This mapping relationship can be obtained by taking the total differential of both sides of the closed-loop vector equations E1x and E1z, as follows:

[0101]

[0102] Simplifying the above formulas, we obtain the mapping relationship between the error of the first passive joint variable in the main chain and the structural error of the first closed-loop sub-chain:

[0103]

[0104] In the formula: δθ1 is the error of the first passive joint variable of the main chain, and δL1,δL4,δd3,δd4,δq1 are the structural parameter errors of the first closed-loop sub-chain.

[0105] This yields the mapping relationship between the error of the first passive joint variable and the error of the first closed-loop subchain structure parameters:

[0106]

[0107] In the formula: δr1=[δL1, δL4, δd3, δd4, δq1]T,

[0108] like Figure 3 The simplified diagram of the second closed-loop subchain structure is shown below. Similarly, the closed-loop constraint equations for the second closed-loop subchain are listed below:

[0109]

[0110] The mapping relationship between the passive joint variable error and the closed-loop structural parameter error of the tandem arm is derived based on the differential method. This mapping relationship can be obtained by taking the total differential of both sides of the closed-loop vector equations E2x and E2z, as follows:

[0111]

[0112] Rearranging the above formulas into matrix form yields the mapping between the error of the second passive joint variable in the main chain and the structural error of the second closed-loop sub-chain:

[0113]

[0114] In the formula: δθ2 is the error of the second passive joint variable of the main chain, and δh1,δd5,δβ0,δd6,δL5,δd2,δq2 are the errors of the structural parameters of the second closed-loop sub-chain.

[0115] This yields the mapping relationship between the second passive joint variable error and the second closed-loop subchain structural parameter error:

[0116]

[0117] In the formula: δr2=[δh1, δd5, δβ0, δd6, δL5, δd2, δq2]T,

[0118] Thus, the error mapping model for the two closed-loop subchains was obtained.

[0119] (3) Establishment and simulation of the overall error mapping model of the robotic arm with closed-loop subchain

[0120] By applying the mapping relationship between passive joint variable errors and structural parameter errors of closed-loop subchains to the error mapping model of the main kinematic chain of the tandem arm, the overall error mapping model can be obtained as follows:

[0121]

[0122] δpose=Jδr

[0123] In the formula: δr = [δh, δL2, δL6, δd1, δd7, δθ0, δr1, δr2]T.

[0124] The above embodiments merely illustrate several implementation methods of this application, and while the descriptions are relatively specific and detailed, they should not be construed as limiting the scope of the invention patent. It should be noted that those skilled in the art can make various modifications and improvements without departing from the concept of this application, and these all fall within the protection scope of this application. Therefore, the protection scope of this patent application should be determined by the appended claims.

[0125] Although embodiments of the invention have been shown and described, it will be understood by those skilled in the art that various changes, modifications, substitutions and alterations can be made to these embodiments without departing from the principles and spirit of the invention, the scope of which is defined by the appended claims and their equivalents.

Claims

1. A step-by-step error modeling and calibration method for serial arms with closed-loop subchains, characterized in that: The steps are as follows: (1) Based on the existing serial robot model with closed-loop subchains, a serial arm model without closed-loop subchains is formed by operation. The kinematics of the model are derived, and then the error calibration model of the serial robot without subchains is established based on the differential motion equations. Then, closed-loop constraint equations are established for the two closed-loop subchains respectively, and the error models of the structural parameter error of the closed-loop subchains and the rotation angle error of the passive joints on the main chain are derived. Finally, the error models of the two closed-loop subchains are added to the previously established error model of the serial robot without subchains, so as to obtain the overall error parameter calibration model of the serial arm with closed-loop subchains. (2) Determine a spiral trajectory within the robot's workspace, with a spiral radius of r and a pitch of h. Verify the feasibility of the trajectory using a robot simulation model, and then conduct a calibration experiment. (3) Set the robot's motion trajectory on the robot control software for the given robot motion trajectory, drive the robot to perform spiral motion, verify that its actual motion trajectory is consistent with the original motion trajectory, and verify the correctness of the model. (4) In order to construct the base coordinate system of the robotic arm, three target ball seats are placed at specific positions on the base plane of the robotic arm. After measuring the position of the target ball on it with a laser tracker, the center point of the three points is taken as the origin of the coordinate system, the direction of the measurement point from the origin to the robotic arm is taken as the x-axis, the upward direction of the normal of the measurement point plane is taken as the z-axis, and the y-axis is determined according to the right-hand rule, thereby constructing a global base coordinate system. (5) Install three target ball seats at the end of the robot, start the robot and execute the robot motion trajectory that has been set, so that the end effector moves along the spiral trajectory, read the joint drive value corresponding to each movement point when the robot end is making spiral motion and monitor the motion state of the robot end in real time, and use a laser tracker to record the spatial position of the three target balls at the end of the trajectory under the base target system. (6) Read the saved experimental data in MATLAB software, process the data to obtain the actual spatial pose of the robot end, and then substitute the driving value corresponding to the trajectory into the forward kinematics equation of the robot to calculate the theoretical spatial pose of the robot end. (7) Based on step (6), in MTATLAB, according to the established error mapping model, the difference between the actual spatial pose and the theoretical spatial pose is calculated to obtain the end-effector pose error. This error is then substituted into the error mapping model to identify the actual structural parameter error of the robot. Subsequently, the identified structural parameter error is added to the structural parameters in the theoretical model to compensate for the robot's motion error. The corrected model is then substituted into the experiment, and the above steps are repeated to verify the accuracy and effectiveness of the identified parameters. The derivation of the main kinematic chain error mapping model is as follows: The main motion chain of this mechanism is a series arm model that forms a series arm without closed-loop sub-chains. It is a three-degree-of-freedom series robotic arm. A fixed Cartesian coordinate system is established in the mechanism. By performing forward kinematics derivation on the main kinematic chains, we obtain the kinematics in a fixed coordinate system. Lower joint drive values ​​converted to real-time pose of the end-effector coordinate system center The relation, and its forward kinematic expression, are as follows: In the formula: s(), c(), t() represent the trigonometric functions sin(), cos(), tan(), respectively; yes abbreviation; The mapping relationship between the pose error and structural parameter error of the tandem arm is derived based on the differential method. This mapping relationship is obtained by taking the total differential of both sides of the forward kinematics equation, as follows: Simplifying the above equations, we obtain the mapping relationship between the end-effector pose error and the kinematic error parameters of the main chain: In the formula: , , , , For end-effector pose error, , , , , , , , This refers to the structural error of the main chain; This establishes the mapping relationship between the end-effector pose error and the main chain structure parameter errors: In the formula: =[ , , , , ] T , =[ , , , , , , , ] T ； The derivation of the closed-loop subchain error mapping model is as follows: When there are errors in the link length and the driven joint variables, the actual passive joint variables are calculated for the closed-loop sub-chain structure, which includes the first closed-loop sub-chain structure and the second closed-loop sub-chain structure. First, let's define the closed-loop constraint equation for the first closed-loop subchain as follows: The mapping relationship between the passive joint variable error and the closed-loop structural parameter error of the tandem arm is derived based on the differential method. This mapping relationship can be obtained by applying the closed-loop vector equation E 1x and E 1z The total differential of both sides is obtained as follows: Simplifying the above formulas, we obtain the mapping relationship between the error of the first passive joint variable in the main chain and the structural error of the first closed-loop sub-chain: In the formula: For the first passive joint variable error of the main chain, , , , , For the structural parameter error of the first closed-loop subchain; This yields the mapping relationship between the error of the first passive joint variable and the error of the first closed-loop subchain structure parameters: In the formula: =[ , , , , ] T J1=- ; Similarly, let's define the closed-loop constraint equations for the second closed-loop subchain: The mapping relationship between the passive joint variable error and the closed-loop structural parameter error of the tandem arm is derived based on the differential method. This mapping relationship can be obtained by applying the closed-loop vector equation E 2x and E 2z The total differential of both sides is obtained as follows: Rearranging the above formulas into matrix form yields the mapping between the error of the second passive joint variable in the main chain and the structural error of the second closed-loop sub-chain: In the formula: For the error of the second passive joint variable in the main chain, For the structural parameter error of the second closed-loop subchain; This yields the mapping relationship between the second passive joint variable error and the second closed-loop subchain structural parameter error: In the formula: =[ ] T J2=- ; Thus, the error mapping model for the two closed-loop subchains was obtained.

2. The step-by-step error modeling and calibration method for serial arms with closed-loop subchains according to claim 1, characterized in that: In step (2), the set spiral trajectory is based on the x-axis direction of the base coordinate system. The spiral radius and pitch are determined according to the robot's workspace, thereby realizing the robot's movement in space.

3. The step-by-step error modeling and calibration method for serial arms with closed-loop subchains according to claim 1, characterized in that: In step (5), the continuous motion of the robot in space is discretized, that is, it needs to be stationary and stable during the motion so that the laser tracker can measure the position when the robot is stable.

4. The step-by-step error modeling and calibration method for serial arms with closed-loop subchains according to claim 1, characterized in that: The method for establishing and simulating the overall error mapping model of a robotic arm with closed-loop subchains is as follows: By applying the mapping relationship between passive joint variable errors and structural parameter errors of closed-loop subchains to the error mapping model of the main kinematic chain of the tandem arm, the overall error mapping model can be obtained as follows: In the formula: =[ ] T .

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