Kinematic calibration method and system for redundant actuated over-constrained parallel robots

Abstract

The application provides a kinematic calibration method and system for a redundant drive over-constrained parallel robot, comprising: constructing a companion error model of each branch of the ROPR by using an exponential product method; describing motion deviation of a redundant joint angle in the error model, and treating deviation of a non-redundant active joint angle as zero; establishing an error feasible space based on a configuration constraint equation; eliminating joint motion error by using a projection method, and establishing an error matrix of the ROPR; measuring end position and posture error of the robot by using a laser tracker; identifying error parameters of the ROPR by using a least square method; and determining drive input of the ROPR according to the identified parameters and performing error compensation. The kinematic calibration method has universality and is easy to be popularized to other redundant drive parallel robots. The kinematic calibration method starts from inherent characteristics of the redundant drive over-constrained parallel robot, and ensures rationality of the error model.

Description

Technical Field

[0001] This invention relates to the technical field of robot calibration, and more specifically, to a kinematic calibration method and system for redundantly driven over-constrained parallel robots. Background Technology

[0002] Compared to non-redundant actuated parallel robots, redundant actuated parallel robots can effectively avoid singularities and reduce joint gap effects, attracting widespread attention from academia and industry. Generally, redundant actuated parallel robots can be divided into two categories: redundant actuated non-over-constrained parallel robots and redundant actuated over-constrained parallel robots. Due to the presence of common or redundant constraints, redundantly actuated over-constrained parallel robots (ROPRs) have advantages such as high stiffness and strong load-bearing capacity, and have promising application prospects.

[0003] Errors are inevitable during robot manufacturing and assembly, such as machining errors in robot parts and assembly errors. These errors result in pose deviations in the final ROPR output end effector, thus reducing the robot's accuracy. To ensure ROPR accuracy, two common techniques are used: improving the precision design of robot parts and employing kinematic calibration techniques for error compensation after assembly. While improving the precision of robot parts can reduce output errors, this method is costly and difficult to implement. Kinematic calibration, on the other hand, involves performing error modeling, measurement, parameter identification, and error compensation on the entire robot after assembly to correct its kinematic inputs, thereby improving the ROPR's output pose accuracy. This approach is superior to improving the precision of robot parts.

[0004] Error modeling is fundamental to kinematic calibration. Currently, there is a wealth of calibration results for non-redundant, non-overconstrained parallel robots. Chinese patent CN113500583B (application number: CN202110760595.0), "A Three-DOF Parallel Robot and Its Calibration Method," uses a closed-loop vector differential method. First, the robot's closed-loop vector equations are established, and then the total differential of these equations is performed to obtain the robot's error model. Chinese patent CN113580148B (application number: CN202111070626.6), "A Kinematic Calibration Method for Parallel Robots Based on Equivalent Kinematic Chains," uses an equivalent kinematic chain method. Grassmann-Cayley algebra is used to construct equivalent virtual serial motion branches for the parallel robot, and then the total differential of its forward kinematic equations is performed to obtain the robot's error matrix.

[0005] Due to the presence of redundant actuation and over-constraint characteristics, common calibration methods for non-redundant, non-over-constraint parallel robots cannot be directly applied to the kinematic calibration of ROPR. Currently, closed-loop vector differentiation and branch-based global error modeling are mainly used for ROPR error modeling. On the one hand, closed-loop vector differentiation lacks universality and ignores the characteristics of over-constraint, which may compromise the rationality of the error model. On the other hand, the branch-based global error model directly ignores the redundant actuation and over-constraint characteristics of ROPR, making it difficult to meet the configuration constraints under geometric errors, thus affecting identification accuracy. Considering these facts, a universal error model that takes into account the characteristics of redundant actuation and over-constraint is still lacking for ROPR. Therefore, it is essential to propose a universal error modeling method that considers redundant actuation and over-constraint characteristics, and on this basis, to form a complete set of kinematic calibration methods applicable to ROPR.

[0006] Therefore, a new technical solution is needed to improve the above-mentioned technical problems. Summary of the Invention

[0007] To address the shortcomings of existing technologies, the purpose of this invention is to provide a kinematic calibration method and system for redundantly driven over-constrained parallel robots.

[0008] According to the present invention, a kinematic calibration method for a redundantly driven over-constrained parallel robot includes the following steps:

[0009] Step S1: Construct the associated error model for each branch of ROPR using the exponential product method;

[0010] Step S2: Describe the motion deviation of redundant joint angles in the error model, and treat the deviation of non-redundant active joint angles as zero;

[0011] Step S3: Establish the error feasible space based on the configuration constraint equations;

[0012] Step S4: Use the projection method to eliminate joint motion errors and establish the ROPR error matrix;

[0013] Step S5: Use a laser tracker to measure the error of the robot's end-effector pose;

[0014] Step S6: Use the least squares method to identify the error parameters of ROPR;

[0015] Step S7: Determine the driving input of ROPR based on the identification parameters and perform error compensation.

[0016] Preferably, in step S1:

[0017] The adjoint error model for each branch of the robot is established based on the exponential product method as follows:

[0018]

[0019] in, Represents the current end pose and initial end pose of branch i; the symbol ∨ indicates mapping se(3) to δη i,j This represents the error of the j-th spiral axis in branch i; k represents the associated error of branch i; st,i This represents the initial terminal error of branch i; The product of the adjoint error matrices; Ad(·) denotes the adjoint transformation of any homogeneous transformation matrix ·; δθ represents the joint angle error of branch i. i,j B represents the angle error of the j-th joint of branch i; i,j The basis for the accompanying error is denoted as .

[0020] Preferably, in step S2:

[0021] The error model describes the motion deviation of redundant joint angles, while treating the deviation of non-redundant active joint angles as zero. The branch error model is described as follows:

[0022] δy i =J e,i k i +J st,i k st,i +J rp,i δθ rp,i ,i=1…n (26)

[0023] Where, δθ rp,i This represents the joint motion deviation of the i-th branch; for non-redundant drive branches, this joint motion deviation only includes passive joint motion deviation; for redundant drive branches, this variable includes both redundant drive active joint motion deviation and passive joint motion deviation.

[0024] Preferably, in step S3:

[0025] Based on the configuration constraints, the geometric error of the robot must fall within the error feasible space; given the positioning type, the configuration constraints are:

[0026]

[0027] Because ROPR has over-constraint properties, J rp The number of rows in the matrix is ​​greater than the number of columns, therefore J e k needs to fall on J rp Only a column space can guarantee that the passive joint has a solution; therefore, the following expression holds:

[0028]

[0029] For J rp Perform singular value decomposition J rp =U∑V T Substituting this into equation (7), we get:

[0030]

[0031] Combining equation (8) for m poses, we get:

[0032]

[0033] in, N represents the position of the i-th pose. e Matrix; this formula indicates that k needs to fall within... On the null space; for Perform QR decomposition, i.e. Substituting into equation (9), the error feasible space is obtained as follows:

[0034]

[0035] Preferably, in step S4:

[0036] The redundant active and passive joint motion deviations in step S2 are eliminated using the projection method, as follows:

[0037]

[0038] Considering that the errors at the ends of each branch are the same, the error models (2) of each branch are integrated as follows:

[0039]

[0040] Among them, δy=δy1=…=δy n Let k be the robot end-effector pose error. Combining step S3, when establishing the error model, the error k needs to fall within the feasible error space, i.e., k = Fe; e represents the coordinate components of error k in the feasible error space; therefore, the ROPR error model is expressed as:

[0041]

[0042] Preferably, in step S5:

[0043] Based on the nominal geometric parameters and actuation values ​​of ROPR, the nominal pose of the end effector is obtained as follows: The end-effector pose of ROPR was determined using a laser tracker as follows:

[0044]

[0045] in, Let be the homogeneous transformation matrix of the end coordinate system {T} relative to the measurement coordinate system {M}. It is the homogeneous transformation matrix of the spatial coordinate system {S} relative to the measurement coordinate system {M};

[0046] The end-effector pose error is calculated based on the nominal end-effector pose and the end-effector measurement pose in equation (14):

[0047]

[0048] Preferably, in step S6:

[0049] The error parameters of ROPR are identified using the least squares method, as follows:

[0050] e r+1 =e r +(J T J) -1 J T δy (36)

[0051] Where r is the iteration number, when ‖δy‖ is less than the set value or e r+1 With e r When the difference is small enough, terminate the iteration.

[0052] Preferably, in step S7:

[0053] The end-effector error obtained in step S6 is compensated into the robot end-effector pose, and the driving input in real time is obtained by inverse ROPR calculation.

[0054] The present invention also provides a kinematic calibration system for a redundantly driven, over-constrained parallel robot, the system comprising the following modules:

[0055] Module M1: Constructs the associated error model for each branch of ROPR using the exponential product method;

[0056] Module M2: Describes the motion deviation of redundant joint angles in the error model, treating the deviation of non-redundant active joint angles as zero;

[0057] Module M3: Establishes the error feasible space based on configuration constraint equations;

[0058] Module M4: Uses projection to eliminate joint motion errors and establishes the ROPR error matrix;

[0059] Module M5: Uses a laser tracker to measure the error of the robot's end-effector pose;

[0060] Module M6: Uses the least squares method to identify the error parameters of ROPR;

[0061] Module M7: Determines the ROPR drive input based on the identification parameters and performs error compensation.

[0062] Preferably, in module M1:

[0063] The adjoint error model for each branch of the robot is established based on the exponential product method as follows:

[0064]

[0065] in, Represents the current end pose and initial end pose of branch i; the symbol ∨ indicates mapping se(3) to δη i,j This represents the error of the j-th spiral axis in branch i; k represents the associated error of branch i; st,i This represents the initial terminal error of branch i; The product of the adjoint error matrices; Ad(·) denotes the adjoint transformation of any homogeneous transformation matrix ·; δθ represents the joint angle error of branch i. i,j B represents the angle error of the j-th joint of branch i; i,j The basis for the accompanying error;

[0066] In module M2:

[0067] The error model describes the motion deviation of redundant joint angles, while treating the deviation of non-redundant active joint angles as zero. The branch error model is described as follows:

[0068] δy i =J e,i k i +J st,i k st,i +J rp,i δθ rp,i ,i=1…n (38)

[0069] Where, δθ rp,i This represents the joint motion deviation of the i-th branch; for non-redundant drive branches, this joint motion deviation only includes passive joint motion deviation; for redundant drive branches, this variable includes both redundant drive active joint motion deviation and passive joint motion deviation.

[0070] In module M3:

[0071] Based on the configuration constraints, the geometric error of the robot must fall within the error feasible space; given the positioning type, the configuration constraints are:

[0072]

[0073] Because ROPR has over-constraint properties, J rp The number of rows in the matrix is ​​greater than the number of columns, therefore J e k needs to fall on J rp Only a column space can guarantee that the passive joint has a solution; therefore, the following expression holds:

[0074]

[0075] For J rp Perform singular value decomposition J rp =U∑V T Substituting this into equation (7), we get:

[0076]

[0077] Combining equation (8) for m poses, we get:

[0078]

[0079] in, N represents the position of the i-th pose. e Matrix; this formula indicates that k needs to fall within... On the null space; for Perform QR decomposition, i.e. Substituting into equation (9), the error feasible space is obtained as follows:

[0080]

[0081] In module M4:

[0082] The projection method is used to eliminate redundant active and passive joint motion deviations in module M2, as follows:

[0083]

[0084] Considering that the errors at the ends of each branch are the same, the error models (2) of each branch are integrated as follows:

[0085]

[0086] Among them, δy=δy1=…=δy n Let k be the robot's end-effector pose error. In conjunction with module M3, when establishing the error model, the error k needs to fall within the feasible error space, i.e., k = Fe; e represents the coordinate components of error k in the feasible error space. Therefore, the ROPR error model is expressed as:

[0087]

[0088] In module M5:

[0089] Based on the nominal geometric parameters and actuation values ​​of ROPR, the nominal pose of the end effector is obtained as follows: The end-effector pose of ROPR was determined using a laser tracker as follows:

[0090]

[0091] in, Let be the homogeneous transformation matrix of the end coordinate system {T} relative to the measurement coordinate system {M}. It is the homogeneous transformation matrix of the spatial coordinate system {S} relative to the measurement coordinate system {M};

[0092] The end-effector pose error is calculated based on the nominal end-effector pose and the end-effector measurement pose in equation (14):

[0093]

[0094] In module M6:

[0095] The error parameters of ROPR are identified using the least squares method, as follows:

[0096] e r+1 =e r +(J T J) -1 J T δy (48)

[0097] Where r is the iteration number, when ‖δy‖ is less than the set value or e r+1 With e r When the difference is small enough, terminate the iteration;

[0098] In module M7:

[0099] The end-effector error obtained from module M6 is compensated into the robot's end-effector pose, and the driving input in real time is obtained through the inverse kinematics calculation of ROPR.

[0100] Compared with the prior art, the present invention has the following beneficial effects:

[0101] 1. The kinematic calibration method of the present invention is universal and can be easily extended to other redundant drive parallel robots;

[0102] 2. The present invention conforms to configuration constraints. The kinematic calibration method of the present invention starts from the inherent characteristics of redundantly driven over-constrained parallel robots, and establishes an error model that considers the characteristics of redundant drive and over-constraint, so that the identified geometric parameters conform to the robot configuration constraints, thus ensuring the rationality of the error model. Attached Figure Description

[0103] Other features, objects, and advantages of the present invention will become more apparent from the following detailed description of non-limiting embodiments with reference to the accompanying drawings:

[0104] Figure 1 This is a flowchart of the kinematic calibration method of the present invention;

[0105] Figure 2 This is a schematic diagram of the redundantly driven over-constrained parallel robot of the present invention;

[0106] Figure 3 This is a schematic diagram illustrating the error measurement principle of the present invention;

[0107] Figure 4 This is a simplified diagram of the 2RRRR-RPS-PSS mechanism according to an embodiment of the present invention;

[0108] Figure 5 This is a diagram showing the positional error of the calibration points before and after the experimental calibration in this embodiment of the invention.

[0109] Figure 6 This is a diagram showing the attitude error of the calibration points before and after the experimental calibration in an embodiment of the present invention.

[0110] Figure 7 This is a diagram showing the positional error of the test points before and after experimental calibration in an embodiment of the present invention.

[0111] Figure 8 This is a diagram showing the attitude error of the test points before and after experimental calibration in an embodiment of the present invention. Detailed Implementation

[0112] The present invention will now be described in detail with reference to specific embodiments. These embodiments will help those skilled in the art to further understand the present invention, but do not limit the invention in any way. It should be noted that those skilled in the art can make several changes and improvements without departing from the concept of the present invention. These all fall within the scope of protection of the present invention.

[0113] Example 1:

[0114] According to the present invention, a kinematic calibration method for a redundantly driven over-constrained parallel robot includes the following steps:

[0115] Step S1: Construct the associated error model for each branch of ROPR using the exponential product method;

[0116] Step S2: Describe the motion deviation of redundant joint angles in the error model, and treat the deviation of non-redundant active joint angles as zero;

[0117] Step S3: Establish the error feasible space based on the configuration constraint equations;

[0118] Step S4: Use the projection method to eliminate joint motion errors and establish the ROPR error matrix;

[0119] Step S5: Use a laser tracker to measure the error of the robot's end-effector pose;

[0120] Step S6: Use the least squares method to identify the error parameters of ROPR;

[0121] Step S7: Determine the driving input of ROPR based on the identification parameters and perform error compensation.

[0122] In step S1:

[0123] The adjoint error model for each branch of the robot is established based on the exponential product method as follows:

[0124]

[0125] in, Represents the current end pose and initial end pose of branch i; the symbol ∨ indicates mapping se(3) to δη i,j This represents the error of the j-th spiral axis in branch i; k represents the associated error of branch i; st,i This represents the initial terminal error of branch i; The product of the adjoint error matrices; Ad(·) denotes the adjoint transformation of any homogeneous transformation matrix ·; δθ represents the joint angle error of branch i. i,j B represents the angle error of the j-th joint of branch i; i,j The basis for the accompanying error is denoted as .

[0126] In step S2:

[0127] The error model describes the motion deviation of redundant joint angles, while treating the deviation of non-redundant active joint angles as zero. The branch error model is described as follows:

[0128] δy i =J e,i k i +J st,i k st,i +J rp,i δθ rp,i ,i=1…n (50)

[0129] Where, δθ rp,iThis represents the joint motion deviation of the i-th branch; for non-redundant drive branches, this joint motion deviation only includes passive joint motion deviation; for redundant drive branches, this variable includes both redundant drive active joint motion deviation and passive joint motion deviation.

[0130] In step S3:

[0131] Based on the configuration constraints, the geometric error of the robot must fall within the error feasible space; given the positioning type, the configuration constraints are:

[0132]

[0133] Because ROPR has over-constraint properties, J rp The number of rows in the matrix is ​​greater than the number of columns, therefore J e k needs to fall on J rp Only a column space can guarantee that the passive joint has a solution; therefore, the following expression holds:

[0134]

[0135] For J rp Perform singular value decomposition J rp =U∑V T Substituting this into equation (7), we get:

[0136]

[0137] Combining equation (8) for m poses, we get:

[0138]

[0139] in, N represents the position of the i-th pose. e Matrix; this formula indicates that k needs to fall within... On the null space; for Perform QR decomposition, i.e. Substituting into equation (9), the error feasible space is obtained as follows:

[0140]

[0141] In step S4:

[0142] The redundant active and passive joint motion deviations in step S2 are eliminated using the projection method, as follows:

[0143]

[0144] Considering that the errors at the ends of each branch are the same, the error models (2) of each branch are integrated as follows:

[0145]

[0146] Among them, δy=δy1=…=δy n Let k be the robot end-effector pose error. Combining step S3, when establishing the error model, the error k needs to fall within the feasible error space, i.e., k = Fe; e represents the coordinate components of error k in the feasible error space; therefore, the ROPR error model is expressed as:

[0147]

[0148] In step S5:

[0149] Based on the nominal geometric parameters and actuation values ​​of ROPR, the nominal pose of the end effector is obtained as follows: The end-effector pose of ROPR was determined using a laser tracker as follows:

[0150]

[0151] in, Let be the homogeneous transformation matrix of the end coordinate system {T} relative to the measurement coordinate system {M}. It is the homogeneous transformation matrix of the spatial coordinate system {S} relative to the measurement coordinate system {M};

[0152] The end-effector pose error is calculated based on the nominal end-effector pose and the end-effector measurement pose in equation (14):

[0153]

[0154] In step S6:

[0155] The error parameters of ROPR are identified using the least squares method, as follows:

[0156] e r+1 =e r +(J T J) -1 J T δy (60)

[0157] Where r is the iteration number, when ‖δy‖ is less than the set value or e r+1 With e r When the difference is small enough, terminate the iteration.

[0158] In step S7:

[0159] The end-effector error obtained in step S6 is compensated into the robot end-effector pose, and the driving input in real time is obtained by inverse ROPR calculation.

[0160] The present invention also provides a kinematic calibration system for a redundantly driven over-constrained parallel robot. The kinematic calibration system for the redundantly driven over-constrained parallel robot can be implemented by executing the process steps of the kinematic calibration method for the redundantly driven over-constrained parallel robot. That is, those skilled in the art can understand the kinematic calibration method for the redundantly driven over-constrained parallel robot as a preferred embodiment of the kinematic calibration system for the redundantly driven over-constrained parallel robot.

[0161] Example 2:

[0162] The present invention also provides a kinematic calibration system for a redundantly driven, over-constrained parallel robot, the system comprising the following modules:

[0163] Module M1: Constructs the associated error model for each branch of ROPR using the exponential product method;

[0164] Module M2: Describes the motion deviation of redundant joint angles in the error model, treating the deviation of non-redundant active joint angles as zero;

[0165] Module M3: Establishes the error feasible space based on configuration constraint equations;

[0166] Module M4: Uses projection to eliminate joint motion errors and establishes the ROPR error matrix;

[0167] Module M5: Uses a laser tracker to measure the error of the robot's end-effector pose;

[0168] Module M6: Uses the least squares method to identify the error parameters of ROPR;

[0169] Module M7: Determines the ROPR drive input based on the identification parameters and performs error compensation.

[0170] In module M1:

[0171] The adjoint error model for each branch of the robot is established based on the exponential product method as follows:

[0172]

[0173] in, Represents the current end pose and initial end pose of branch i; the symbol ∨ indicates mapping se(3) to δη i,j This represents the error of the j-th spiral axis in branch i; k represents the associated error of branch i; st,i This represents the initial terminal error of branch i; The product of the adjoint error matrices; Ad(·) denotes the adjoint transformation of any homogeneous transformation matrix ·; δθ represents the joint angle error of branch i. i,j B represents the angle error of the j-th joint of branch i; i,j The basis for the accompanying error;

[0174] In module M2:

[0175] The error model describes the motion deviation of redundant joint angles, while treating the deviation of non-redundant active joint angles as zero. The branch error model is described as follows:

[0176] δy i =J e,i k i +J st,i k st,i +J rp,i δθ rp,i ,i=1…n (62)

[0177] Where, δθ rp,i This represents the joint motion deviation of the i-th branch; for non-redundant drive branches, this joint motion deviation only includes passive joint motion deviation; for redundant drive branches, this variable includes both redundant drive active joint motion deviation and passive joint motion deviation.

[0178] In module M3:

[0179] Based on the configuration constraints, the geometric error of the robot must fall within the error feasible space; given the positioning type, the configuration constraints are:

[0180]

[0181] Because ROPR has over-constraint properties, J rp The number of rows in the matrix is ​​greater than the number of columns, therefore J e k needs to fall on J rp Only a column space can guarantee that the passive joint has a solution; therefore, the following expression holds:

[0182]

[0183] For J rp Perform singular value decomposition J rp =U∑V T Substituting this into equation (7), we get:

[0184]

[0185] Combining equation (8) for m poses, we get:

[0186]

[0187] in, N represents the position of the i-th pose. e Matrix; this formula indicates that k needs to fall within... On the null space; for Perform QR decomposition, i.e. Substituting into equation (9), the error feasible space is obtained as follows:

[0188]

[0189] In module M4:

[0190] The projection method is used to eliminate redundant active and passive joint motion deviations in module M2, as follows:

[0191]

[0192] Considering that the errors at the ends of each branch are the same, the error models (2) of each branch are integrated as follows:

[0193]

[0194] Among them, δy=δy1=…=δy n Let k be the robot's end-effector pose error. In conjunction with module M3, when establishing the error model, the error k needs to fall within the feasible error space, i.e., k = Fe; e represents the coordinate components of error k in the feasible error space. Therefore, the ROPR error model is expressed as:

[0195]

[0196] In module M5:

[0197] Based on the nominal geometric parameters and actuation values ​​of ROPR, the nominal pose of the end effector is obtained as follows: The end-effector pose of ROPR was determined using a laser tracker as follows:

[0198]

[0199] in, Let be the homogeneous transformation matrix of the end coordinate system {T} relative to the measurement coordinate system {M}. It is the homogeneous transformation matrix of the spatial coordinate system {S} relative to the measurement coordinate system {M};

[0200] The end-effector pose error is calculated based on the nominal end-effector pose and the end-effector measurement pose in equation (14):

[0201]

[0202] In module M6:

[0203] The error parameters of ROPR are identified using the least squares method, as follows:

[0204] e r+1 =e r +(J T J) -1 J T δy (72)

[0205] Where r is the iteration number, when ‖δy‖ is less than the set value or e r+1 With e r When the difference is small enough, terminate the iteration;

[0206] In module M7:

[0207] The end-effector error obtained from module M6 is compensated into the robot's end-effector pose, and the driving input in real time is obtained through the inverse kinematics calculation of ROPR.

[0208] Example 3:

[0209] First, an adjoint error model for each branch is established based on the exponential product method. To address the impact of redundant drive characteristics, deviations in the joint angles of redundant drives are described in both the error model and the forward kinematics of each iteration. To address the impact of over-constraint characteristics, an error feasible space is established based on the error model to constrain the identified errors. Based on this, an error model for the robot is established, which theoretically conforms to the robot's configuration constraints. Next, a laser tracker is used to measure the robot's end-effector pose and determine the errors, and the error parameters are identified using the least squares method. Finally, the identified parameters are substituted into the ROPR inverse kinematics to determine the ROPR drive inputs containing errors, and real-time error compensation is performed on the ROPR.

[0210] Specifically, the present invention consists of the following seven steps (e.g. Figure 1 ):

[0211] Step 1: Construct the associated error model for each branch of ROPR using the exponential product method.

[0212] like Figure 2 As shown, ROPR has n branches, each with degrees of freedom fi, and its end-effector moving platform has degrees of freedom f. For branch i, its forward kinematics can be expressed as:

[0213]

[0214] in, Represents the current end pose and initial end pose of branch i; θ i,j ,j∈1,…,f i Let ξ be the j-th joint variable of branch i;i,j This is the representation of the j-th spiral axis of branch i in the spatial coordinate system. The symbol ∧ indicates that... Mapped to The symbol ∨ represents mapping se(3) to... By performing a first-order variational analysis on equation (73), the error model for branch i can be obtained as follows:

[0215]

[0216] Where, δη i,j This represents the error of the j-th spiral axis in branch i; k represents the associated error of branch i; st,i This represents the initial terminal error of branch i; The product of the adjoint error matrices; Ad(·) denotes the adjoint transformation of any homogeneous transformation matrix ·; δθ represents the joint angle error of branch i. i,j B represents the angle error of the j-th joint of branch i; i,j As the basis for the accompanying error. For the helical axis corresponding to the rotary joint. Its accompanying error basis is:

[0217]

[0218] Where ω i,j ,m i,j ,n i,j These are mutually perpendicular unit vectors. For the helical axis corresponding to the moving joint... Its accompanying error basis is:

[0219]

[0220] Where v i,j ,m i,j ,n i,j These are mutually perpendicular unit vectors.

[0221] Step 2: Describe the motion deviation of redundant joint angles in the error model, while treating the deviation of non-redundant active joint angles as zero.

[0222] Considering that the redundant active joint angles of the ROPR are determined by the remaining non-redundant active joint angles, there will be a deviation in the redundant active joint angles in the presence of geometric parameter errors. This deviation changes with the configuration. Meanwhile, the deviation of the non-redundant active joint angles is mainly caused by the steady-state error of the controller and can be ignored. For a ROPR with f degrees of freedom, it can be assumed that its branches from f+1 to n are redundant drive branches, and the first joint is an active joint. Therefore, equation (74) can be further written as:

[0223] δyi =J e,i k i +J st,i k st,i +J rp,i δθ rp,i ,i=1…n (77)

[0224] in, The Jacobian matrix represents the joint motion deviation. It includes redundant active joint angles and passive joint angle motion deviations.

[0225] Step 3: Establish the error feasible space based on the configuration constraint equations.

[0226] Since all branches share the same end-effector platform, the end-effector errors of each branch are the same, i.e., δy1=…δy n =δy,k st,1 =…=k st,n =k st Therefore, the constraint equations can be obtained:

[0227]

[0228] Because ROPR has over-constraint properties, J rp The number of rows in the matrix is ​​greater than the number of columns, therefore J e k needs to fall on J rp Only a column space can guarantee that the passive joint has a solution. Therefore, the following expression holds:

[0229]

[0230] For J rp Perform singular value decomposition J rp =U∑V T Substituting this into equation (79), we get:

[0231]

[0232] Combining equation (80) for m poses, we get:

[0233]

[0234] in, N represents the position of the i-th pose. e Matrix. This formula shows that k needs to fall within... On the null space. For Perform QR decomposition, i.e. Substituting into equation (81), the feasible error space can be obtained as follows:

[0235]

[0236] Step 4: Use the projection method to eliminate joint motion deviations and establish the ROPR error matrix.

[0237] By using the projection method to eliminate redundant active joint and passive joint motion deviations in equation (74), the equation can be simplified to:

[0238]

[0239] Where N rp,i For J rp,i The left null space. Considering that the errors at the ends of each branch are the same, the error models (83) of each branch can be integrated as follows:

[0240]

[0241] Among them, δy=δy1=…=δy n Let be the robot end-effector pose error. To ensure that the identified error falls within the error model, when establishing the error model, the error k needs to fall within the error feasible space, i.e., k = Fe. e represents the coordinate components of the error k in the error feasible space. Combining equation (84), the error model of ROPR can be expressed as:

[0242]

[0243] Step 5: Use a laser tracker to measure the error of the end effector pose of the parallel robot.

[0244] This step involves determining the nominal pose of the ROPR end effector in the spatial coordinate system before identifying error parameters, and then using a laser tracker to determine the measured pose of the end effector, such as... Figure 3 As shown.

[0245] First, based on the nominal geometric parameters and actuation values ​​of ROPR, the nominal pose of the end effector can be obtained as follows: The end-effector pose of ROPR can be determined using a laser tracker as follows:

[0246]

[0247] in, Let be the homogeneous transformation matrix of the end coordinate system {T} relative to the measurement coordinate system {M}. It is the homogeneous transformation matrix of the spatial coordinate system {S} relative to the measurement coordinate system {M}.

[0248] The end-effector pose error is calculated based on the nominal end-effector pose and the end-effector measurement pose in equation (86):

[0249]

[0250] Step 6: Use the least squares method to identify the error parameters of ROPR.

[0251] Based on equation (85), the error identification of ROPR is transformed into a nonlinear least squares problem with an iterative relationship, as follows:

[0252] e r+1 =e r +(J T J) -1 J T δy (88)

[0253] Where r is the iteration number, when ‖δy‖ is less than the set value or e r+1 With e r When the difference is small enough, terminate the iteration.

[0254] Step 7: Determine the driving input of ROPR based on the identification parameters and perform error compensation.

[0255] The obtained end-effector error is compensated into the robot end-effector pose, and the driving input in real time is obtained through the inverse kinematics of ROPR, thereby improving the accuracy of the robot end-effector output.

[0256] Calibration of a 2RRRR-RPS-PSS redundantly driven over-constrained parallel robot: The 2RRRR-RPS-PSS mechanism is a redundantly driven over-constrained parallel mechanism with two rotations and one transfer. Figure 4 (This is a simplified diagram of the mechanism), which consists of a moving platform, a fixed platform, two RRRR branches, one RPS branch, and one PSS branch. Each branch is located at B. i and C i Point O is connected to the fixed base and the end effector moving platform. A spatial coordinate system {S} is connected to the fixed base located at point O, where the z-axis is perpendicular to the plane A1A2A3A4, the x-axis points from A1 to A3, and the y-axis follows the right-hand rule. The tool frame {T} is connected to the end effector moving platform located at the center of the end effector surface E2, where the u-axis is parallel to C1C3, the v-axis runs from C2 to C4, and the w-axis follows the right-hand rule. Its link parameters are defined as follows: B i C i =l, i=1,…,4, OA1=OA3=l1, OA2=OA4=l2, O′C1=O′C3=l3, E1C2=E1C4=l4, C1D1=C3D3=l5, O′E1=h1, E1E2=h2.

[0257] Since the ROPR can translate along the z-axis and rotate about the u and y-axis, three generalized coordinates [z, α, β] are defined to describe the motion of the mobile platform about {S}, where z represents the z-coordinate of O′ about {S}; α and β represent the rotation angles about the u and y axes, respectively. Given the generalized coordinates, the orientation of {T} in {S} can be expressed as:

[0258]

[0259] Where s x =[1,0,0] T ,s y =[0,1,0] T ,s z =[0,0,1] T yes The standard unit vector in the vector; s and c represent the sine and cosine functions, respectively. From this, the coordinates of all joints of the mechanism can be calculated. B i The coordinates of points i = 1…4 are Where qi is the driving quantity. Point C i The coordinates of i = 1…4 are defined as D i The coordinates are

[0260] The joint motion spirals on branch 1 can be represented as:

[0261]

[0262] The joint motion spirals on branch 2 can be represented as:

[0263]

[0264] The joint motion spirals on branch 3 can be represented as:

[0265]

[0266] The joint motion spirals on branch 4 can be represented as:

[0267]

[0268] According to equation (74), the adjoint error model of branch 1 can be calculated as follows:

[0269]

[0270] Where k1 is the geometric error vector of branch 1; k st,1Let δθ1 be the initial end-effector error of branch 1; δθ1 be the joint motion deviation of branch 1. Similarly, the adjoint error models for branches 2 to 4 can be obtained as follows:

[0271]

[0272]

[0273]

[0274] Where k2, k3, and k4 are the geometric error vectors of branches 2, 3, and 4, respectively; k st,2 ,k st,3 ,k st,4 δθ2, δθ3, and δθ4 represent the initial end-effector errors of branches 2, 3, and 4, respectively; δθ2, δθ3, and δθ4 represent the joint motion deviations of branches 2, 3, and 4.

[0275] Considering the redundant active joint angle θ of the mechanism 4,1 There is a deviation; the remaining active joint motion deviations are δθ 1,1 ,δθ 2,1 ,δθ 3,1 It can be treated as zero, i.e., δθ 1,1 =δθ 2,1 =δθ 3,1 =0. In equations (94)-(95), δθ 1,1 ,δθ 2,1 ,δθ 3,1 The relevant columns can be further eliminated, and the result can be sorted as follows:

[0276]

[0277]

[0278]

[0279]

[0280] Among them, J rp,i ,i=1…4 represents the Jacobian matrix of joint motion deviations; δθ rp,i ,i=1…4 represents the joint angle motion deviation.

[0281] Eliminating δθ in equation (96) using the projection method rp,1 ,δθ rp,2 ,δθ rp,3 ,δθ rp,4 The expression can be simplified to:

[0282]

[0283]

[0284]

[0285]

[0286] Where N rp,i For J rp,i The left null space, i.e.

[0287] Furthermore, the feasible error space containing the geometric error is derived. Considering that all branches share the same end-effector moving platform, the end-effector errors of each branch are the same, i.e., δy1=…δy4=δy,k st,1 =…=k st,4 =k st Combining this with equation (96), we can obtain the constraint equation:

[0288]

[0289] in Therefore, this equation relates to δθ rp Over-constraint, J e k needs to fall on J rp Only a column space can guarantee that the passive joint has a solution. Therefore, the following expression holds:

[0290]

[0291] For J rp Perform singular value decomposition J rp =U∑V T Substituting this into equation (79), we get:

[0292]

[0293] Combining equation (80) for m poses, we get:

[0294]

[0295] in, N represents the position of the i-th pose. e Matrix. This formula shows that k needs to fall within... On the null space. For Perform QR decomposition, i.e. Substituting into equation (81), the feasible error space can be obtained as follows:

[0296]

[0297] Substituting equations (102) and (97) into equation (85), we obtain the error model of the mechanism as follows:

[0298]

[0299] in:

[0300]

[0301]

[0302] Based on the measurement principle, calibration simulation was performed on the 2RRRR-RPS-PSS redundant driven over-constrained parallel robot, and the parameters of the mechanism are shown in Table 1.

[0303]

[0304] By uniformly sampling the workspace α∈[-25,25]deg×β∈[-25,25]deg×z∈[300,600]mm, 97 calibration points were obtained for error identification. In addition, 72 test points were randomly generated in the workspace to verify the accuracy of the identification results. 78 geometric errors e were calculated according to a normal distribution N(1×10⁻⁶). -3 1×10 -6 Random selection. Considering the laser tracker's measurement accuracy is 5μm, the random error N(5×10⁻⁶) follows a normal distribution. -5 2.5×10 -9 ) is added to the end measurement location, while obeying N(5×10 -6 2.5×10 -11 Random errors are added to the end-effector's measured posture. Based on a kinematic calibration method for a redundantly driven over-constrained parallel robot, the end-effector pose is determined using the identification results, and the drive joint drive values ​​are recalculated. The compensated drive values ​​are then used to drive the mechanism through a compensation experiment to complete the kinematic calibration.

[0305] For the calibration point, the positional error before and after calibration is as follows: Figure 5 As shown, the average and maximum values ​​of the positional error, as well as the standard deviation, changed from the original 0.0042m, 0.006m, and 8.35×10, respectively. -4 m decreased to 0.000078m, 0.00018m, and 4.39×10 -5 m. The attitude error before and after calibration is as follows: Figure 6 As shown, the average and maximum values ​​of the attitude error, as well as the standard deviation, decreased from 0.0077 rad, 0.0178 rad, and 0.0043 rad to 0.00044 rad, 0.00119 rad, and 0.00027 rad, respectively.

[0306] For the test point, the positional error before and after calibration is as follows: Figure 7As shown, the average and maximum values ​​of the position error, as well as the standard deviation, decreased from 0.00434m, 0.00568m, and 0.00068m to 0.00008m, 0.00016m, and 0.00003m, respectively. The attitude errors before and after calibration are shown below. Figure 8 As shown, the average and maximum values ​​of the attitude error, as well as the standard deviation, decreased from 0.0067 rad, 0.0159 rad, and 0.0040 rad to 0.00045 rad, 0.00084 rad, and 0.00018 rad, respectively.

[0307] The results show that the calibration effect is significant, verifying the correctness of the error model.

[0308] Those skilled in the art can understand this embodiment as a more specific description of Embodiment 1 and Embodiment 2.

[0309] Those skilled in the art will understand that, besides implementing the system and its various devices, modules, and units provided by this invention in the form of purely computer-readable program code, the same functions can be achieved entirely through logical programming of the method steps, making the system and its various devices, modules, and units of this invention function in the form of logic gates, switches, application-specific integrated circuits, programmable logic controllers, and embedded microcontrollers. Therefore, the system and its various devices, modules, and units provided by this invention can be considered as a hardware component, and the devices, modules, and units included therein for implementing various functions can also be considered as structures within the hardware component; alternatively, the devices, modules, and units for implementing various functions can be considered as both software modules implementing the method and structures within the hardware component.

[0310] Specific embodiments of the present invention have been described above. It should be understood that the present invention is not limited to the specific embodiments described above, and those skilled in the art can make various changes or modifications within the scope of the claims, which do not affect the essence of the present invention. Unless otherwise specified, the embodiments and features described in this application can be arbitrarily combined with each other.

Claims

1. A kinematic calibration method for a redundantly driven, over-constrained parallel robot, characterized in that, The method includes the following steps: Step S1: Construct the associated error model for each branch of ROPR using the exponential product method; Step S2: Describe the motion deviation of redundant joint angles in the error model, and treat the deviation of non-redundant active joint angles as zero; Step S3: Establish the error feasible space based on the configuration constraint equations; Step S4: Use the projection method to eliminate joint motion errors and establish the ROPR error matrix; Step S5: Use a laser tracker to measure the error of the robot's end-effector pose; Step S6: Use the least squares method to identify the error parameters of ROPR; Step S7: Determine the driving input of ROPR based on the identification parameters and perform error compensation.

2. The kinematic calibration method for a redundantly driven over-constrained parallel robot according to claim 1, characterized in that, In step S1: The adjoint error model for each branch of the robot is established based on the exponential product method as follows: Among them, g i,st , Represents the current end pose and initial end pose of branch i; the symbol ∨ indicates mapping se(3) to δη i,j This represents the error of the j-th spiral axis in branch i; k represents the associated error of branch i; st, This represents the initial terminal error of branch i; The product of the adjoint error matrices; Ad(·) denotes the adjoint transformation of any homogeneous transformation matrix ·; δθ represents the joint angle error of branch i. i,j B represents the angle error of the j-th joint of branch i; i,j The basis for the accompanying error is denoted as .

3. The kinematic calibration method for a redundantly driven over-constrained parallel robot according to claim 1, characterized in that, In step S2: The error model describes the motion deviation of redundant joint angles, while treating the deviation of non-redundant active joint angles as zero. The branch error model is described as follows: δy i =J e,i k i +J st,i k st,i +J rp,i δθ rp,i ,i=1…n (2) Where, δθ rp, This represents the joint motion deviation of the i-th branch; for non-redundant drive branches, this joint motion deviation only includes passive joint motion deviation; for redundant drive branches, this variable includes both redundant drive active joint motion deviation and passive joint motion deviation.

4. The kinematic calibration method for a redundantly driven over-constrained parallel robot according to claim 1, characterized in that, In step S3: Based on the configuration constraints, the geometric error of the robot must fall within the error feasible space; given the positioning type, the configuration constraints are: Because ROPR has over-constraint properties, J rp The number of rows in the matrix is ​​greater than the number of columns, therefore J e k needs to fall on J rp Only a column space can guarantee that the passive joint has a solution; therefore, the following expression holds: For J rP Perform singular value decomposition j rP =∑V T Substituting this into equation (7), we get: Combining equation (8) for m poses, we get: in, N represents the position of the i-th pose. e Matrix; this formula indicates that k needs to fall within... On the null space; for Perform QR decomposition, i.e. Substituting into equation (9), the error feasible space is obtained as follows:

5. The kinematic calibration method for a redundantly driven over-constrained parallel robot according to claim 1, characterized in that, In step S4: The redundant active and passive joint motion deviations in step S2 are eliminated using the projection method, as follows: Considering that the errors at the ends of each branch are the same, the error models (2) of each branch are integrated as follows: Among them, δy=δy1=…=δy n Let k be the robot end-effector pose error. Combining step S3, when establishing the error model, the error k needs to fall within the feasible error space, i.e., k = Fe; e represents the coordinate components of error k in the feasible error space; therefore, the ROPR error model is expressed as:

6. The kinematic calibration method for a redundantly driven over-constrained parallel robot according to claim 1, characterized in that, In step S5: Based on the nominal geometric parameters and actuation values ​​of ROPR, the nominal pose of the end effector is obtained as follows: The end-effector pose of ROPR was determined using a laser tracker as follows: in, Let be the homogeneous transformation matrix of the end coordinate system {T} relative to the measurement coordinate system {M}. It is the homogeneous transformation matrix of the spatial coordinate system {S} relative to the measurement coordinate system {M}; The end-effector pose error is calculated based on the nominal end-effector pose and the end-effector measurement pose in equation (14):

7. The kinematic calibration method for a redundantly driven over-constrained parallel robot according to claim 1, characterized in that, In step S6: The error parameters of ROPR are identified using the least squares method, as follows: e r+1 e r +(J t J) -1 J T δY ( 12 Where r is the iteration number, when |δY| is less than the set value or E r+1 With e r When the difference is small enough, terminate the iteration.

8. The kinematic calibration method for a redundantly driven over-constrained parallel robot according to claim 1, characterized in that, In step S7: The end-effector error obtained in step S6 is compensated into the robot end-effector pose, and the driving input in real time is obtained by inverse ROPR calculation.

9. A kinematic calibration system for a redundantly driven, over-constrained parallel robot, characterized in that, The system includes the following modules: Module M1: Constructs the associated error model for each branch of ROPR using the exponential product method; Module M2: Describes the motion deviation of redundant joint angles in the error model, treating the deviation of non-redundant active joint angles as zero; Module M3: Establishes the error feasible space based on configuration constraint equations; Module M4: Uses projection to eliminate joint motion errors and establishes the ROPR error matrix; Module M5: Uses a laser tracker to measure the error of the robot's end-effector pose; Module M6: Uses the least squares method to identify the error parameters of ROPR; Module M7: Determines the ROPR drive input based on the identification parameters and performs error compensation.

10. The kinematic calibration system for a redundantly driven over-constrained parallel robot according to claim 9, characterized in that, In module M1: The adjoint error model for each branch of the robot is established based on the exponential product method as follows: Among them, g i,st , Represents the current end pose and initial end pose of branch i; the symbol ∨ indicates mapping se(3) to δη i,j This represents the error of the j-th spiral axis in branch i; k represents the associated error of branch i; st, This represents the initial terminal error of branch i; The product of the adjoint error matrices; Ad(·) denotes the adjoint transformation of any homogeneous transformation matrix ·; δθ represents the joint angle error of branch i. i,j B represents the angle error of the j-th joint of branch i; i,j The basis for the accompanying error; In module M2: The error model describes the motion deviation of redundant joint angles, while treating the deviation of non-redundant active joint angles as zero. The branch error model is described as follows: δy i =J e,i k i +J st,i k st,i +J rp,i δθ rp,i ,i=1…n (14) Where, δθ rp, This represents the joint motion deviation of the i-th branch; for non-redundant drive branches, this joint motion deviation only includes passive joint motion deviation; for redundant drive branches, this variable includes both redundant drive active joint motion deviation and passive joint motion deviation. In module M3: Based on the configuration constraints, the geometric error of the robot must fall within the error feasible space; given the positioning type, the configuration constraints are: Because ROPR has over-constraint properties, J rp The number of rows in the matrix is ​​greater than the number of columns, therefore J e k needs to fall on J rp Only a column space can guarantee that the passive joint has a solution; therefore, the following expression holds: For J rp Perform singular value decomposition J rp =∑V T Substituting this into equation (7), we get: Combining equation (8) for m poses, we get: in, N represents the position of the i-th pose. e Matrix; this formula indicates that k needs to fall within... On the null space; for Perform QR decomposition, i.e. Substituting into equation (9), the error feasible space is obtained as follows: In module M4: The projection method is used to eliminate redundant active and passive joint motion deviations in module M2, as follows: Considering that the errors at the ends of each branch are the same, the error models (2) of each branch are integrated as follows: Among them, δy=δy1=…=δy n Let k be the robot's end-effector pose error. In conjunction with module M3, when establishing the error model, the error k needs to fall within the feasible error space, i.e., k = Fe; e represents the coordinate components of error k in the feasible error space. Therefore, the ROPR error model is expressed as: In module M5: Based on the nominal geometric parameters and actuation values ​​of ROPR, the nominal pose of the end effector is obtained as follows: The end-effector pose of ROPR was determined using a laser tracker as follows: in, Let be the homogeneous transformation matrix of the end coordinate system {T} relative to the measurement coordinate system {M}. It is the homogeneous transformation matrix of the spatial coordinate system {S} relative to the measurement coordinate system {M}; The end-effector pose error is calculated based on the nominal end-effector pose and the end-effector measurement pose in equation (14): In module M6: The error parameters of ROPR are identified using the least squares method, as follows: e r+1 e r +(J T J) -1 J T δy ( 24 Where r is the iteration number, when ‖δy‖ is less than the set value or e r+1 With e r When the difference is small enough, terminate the iteration; In module M7: The end-effector error obtained from module M6 is compensated into the robot's end-effector pose, and the driving input in real time is obtained through the inverse kinematics calculation of ROPR.

Images