Method for improving positioning precision of redundant mechanical arm based on null-space stiffness optimization

Abstract

The invention provides a method for improving the positioning precision of a redundant mechanical arm based on null-space stiffness optimization. According to the method, firstly, a static stiffness model of the redundant mechanical arm needs to be constructed, parameter calibration is conducted on a joint stiffness coefficient of the mechanical arm based on the static stiffness model, after the joint stiffness coefficient of the mechanical arm is obtained, a stiffness performance index of the mechanical arm is defined, and finally the positioning precision of the redundant mechanical arm is improved based on a null-space stiffness optimization method. In conclusion, the method for improving the positioning precision of the redundant mechanical arm based on null-space stiffness optimization provides a new and effective solution for improving the precision of the mechanical arm, and the optimal group of joint configurations can be selected from the numerous groups of joint configurations of the redundant mechanical arm to perform production operation, so that the production quality of the operation is improved; the method has a wide application prospect in the fields of precision manufacturing and the like.

Description

Technical Field

[0001] The present invention belongs to the field of methods for improving positioning accuracy of mechanical arms, and in particular relates to a method for improving positioning accuracy of redundant mechanical arms based on zero space stiffness optimization. Background Art

[0002] With the rapid development of science and technology and industry, the application of robotic arms in manufacturing, aerospace, medical, service robots and other fields is becoming more and more extensive. Especially in high-precision operation scenarios, such as precision assembly, micro-nano operation, space exploration, etc., extremely high requirements are placed on the positioning accuracy of the robotic arm. Redundant robotic arms have become a research hotspot in these fields because of their higher flexibility and fault tolerance. However, the complexity and multi-degree-of-freedom characteristics of redundant robotic arms have brought great challenges to the control of positioning accuracy. As the number of degrees of freedom of the robotic arm increases, its weak stiffness characteristics are more significant. This makes it difficult for the positioning accuracy of the robotic arm to reach the ideal state under complex working conditions, which seriously affects the operating efficiency and product quality. In view of the weak stiffness problem of the current redundant robotic arm, it is urgent to propose a robotic arm stiffness performance index to measure its mechanical characteristics, and at the same time, optimize the configuration of the robotic arm according to the stiffness performance index to improve the stiffness performance of the robotic arm, so as to meet higher standard operation requirements. Summary of the invention

[0003] The present invention proposes a method for improving the positioning accuracy of redundant manipulators based on zero-space stiffness optimization, which provides a new and effective solution for improving the accuracy of manipulators. The optimal set of joint configurations of the redundant manipulators can be selected from the countless sets of joint configurations for production operations to improve the production quality of the operations. This method has broad application prospects in the fields of precision manufacturing and the like. Specifically, it includes the following four main steps, as shown in Figure 1: Step 1: Construct the static stiffness model of the robot arm; Step 2: Calibrate the joint stiffness of the robot arm; Step 3: Define the stiffness characteristic index of the robot arm; Step 4: Optimize the redundant robot arm stiffness based on zero space.

[0004] The various steps of the method for improving the positioning accuracy of redundant manipulators based on zero space stiffness optimization are described as follows: Step 1: Construct the static stiffness model of the robot arm: Reference Figure 1 As shown, according to the forward kinematics of the robot arm, the robot arm end coordinate system can be established Relative to the robot base coordinate system The posture relationship between them is: (1) in is the forward motion equation of the robot arm, is the number of degrees of freedom of the robot arm, Indicates The rotation angle of each joint, Represents the homogeneous transformation matrix of the end of the robot arm relative to the robot arm base. Differentiating formula (1), we can obtain the following relationship: (2) in is the deformation of the robot joint, The position and shape variable vector of the end of the robot arm relative to the base can be converted into a homogeneous transformation matrix: (3) in To convert a vector into a homogeneous transformation matrix function, is the position deformation of the origin of the robot arm end coordinate system relative to the robot arm base coordinate system, is the posture deformation of the robot end coordinate system relative to the robot base coordinate system, where The specific expression is as follows: (4) (5) (6) in , and Respectively represent the coordinate system of the end of the robot arm relative to the coordinate system of the robot arm base and The rotational deformation of the axis. Jacobian matrix ,in Representative The specific expression is: (7) in For the coordinate system { o i x i y i z i}of The direction vector of the axis relative to the machine base coordinate system, Represents the coordinate system { o i x i y i z i}The attitude matrix relative to the base coordinate system, is the position of the end of the robot relative to the coordinate system { o i x i y i z i} position; Due to the weak stiffness characteristics of the robotic arm, the relationship between the deformation of the end of the robotic arm and the deformation of the end load can be determined based on Hooke's law: (8) in is the Cartesian joint stiffness matrix, is the spatial force vector applied to the end of the robotic arm, which is composed of the spatial force and torque At the same time, each joint of the robot arm can be equivalent to a linear spring, refer to Figure 2 Therefore, the relationship between the joint torque and the joint deformation can be obtained as follows: (9) in is the joint stiffness matrix, is the joint torque. According to the principle of virtual work, the mapping relationship between the end load of the robotic arm and the joint torque of the robotic arm can be constructed as follows: (10) Apply formula (10) to By differentiating, we can obtain the following relationship: (11) Substituting formulas (2), (8) and (9) into formula (11), we can obtain: (12) in To supplement the stiffness matrix, the specific expression is: (13) According to formula (12), the mapping relationship between the Cartesian stiffness matrix and the joint stiffness matrix can be determined as: (14) Usually the supplementary stiffness matrix can be ignored, so formula (14) can be further simplified as: (15) Substituting formula (15) into formula (8), the corresponding static stiffness model can be obtained as follows: (16) Step 2: Calibration of the joint stiffness of the robotic arm: After the static stiffness model of the manipulator is constructed, the stiffness coefficient of the manipulator can be calibrated based on the model. In order to avoid solving the inverse of the Jacobian matrix, formula (16) can be rewritten as: (17) For formula (17) By separating the variables, we can obtain the following form: (18) in is the parameter to be calibrated, the matrix The expression is: (19) Finally, the joint stiffness calibration of the robot arm can be transformed into an optimization problem to minimize the positioning error of the robot end: (20) in represents the function to be optimized, r represents the number of measurement points sampled, Indicates j The deformation variables of the end pose of each sampling point are minimized to achieve the purpose of calibrating the stiffness of the robot joints; Step 3: Define the stiffness characteristic index of the robot arm: After obtaining the joint stiffness of the robot arm, the stiffness performance evaluation index of the robot arm is defined according to the joint stiffness. Based on formula (17), , it can be proved . For the matrix Blocking can be decomposed into 4 3 3 matrix submodules, namely: (21) in is the displacement stiffness submatrix, is the rotational stiffness submatrix, is the mixed stiffness sub-matrix. Usually the end of the robot arm is only subject to spatial forces The effect of the torque is mainly to produce deformation in the position direction of the end of the robot arm. can be ignored, and the following relationship can be obtained: (22) When the robot is in a certain position When a unit space force is applied to the end of the robot arm , the deformation at the end is related to the direction of the force. According to formula (22), the square of the deformation at the end position can be obtained as: (23) in is the deformation of the end position of the robot under unit spatial force, and the spatial force The following relations are satisfied: (24) In order to solve the maximum deformation caused by unit force, the Lagrange equation is constructed. The specific expression is: (25) in is the Lagrange equation, is the Lagrange multiplier, and the partial derivative of formula (25) is: (26) According to formula (26), For the matrix Substituting formula (26) into formula (23), we can obtain the following relationship: (27) Defining the Matrix The eigenvalue of ,Right now ， , where the eigenvalue and The corresponding eigenvectors are , and . Therefore, the following relationship can be obtained: (28) According to formulas (27) and (28), it can be shown that a unit force is applied to the end of the robot arm along and The directions produce the maximum and minimum deformations, respectively, which also means that the greater the deformation, the lower the stiffness in that direction. Then, when the robot arm is in a certain position, the matrix The corresponding maximum eigenvalue is the stiffness performance index of the robot arm; Step 4: Redundant manipulator stiffness optimization based on null space: After defining the stiffness characteristic index of the robot arm, this patent optimizes the stiffness of the redundant robot arm. The inverse kinematics of the redundant robot arm usually corresponds to an infinite number of sets of solutions, but there is an optimal set of stiffness configurations among these infinite configurations. The patent of this invention adopts the method of zero space optimization to obtain the optimal stiffness configuration of the robot arm. By introducing the concept of zero space, the motion space of the robot arm is decomposed into two parts: task space and zero space. In the task space, the robot arm completes the main task according to the predetermined trajectory and speed; in the zero space, the redundant degrees of freedom are used to optimize the internal characteristics of the robot arm, thereby improving the overall positioning accuracy of the robot arm. For formula (2), the corresponding general solution expression is: (29) in represents the generalized joint velocity, is the Jacobian matrix The generalized inverse of is solved as follows: (30) is the null space projection matrix, which is expressed as follows: (31) in is the unit matrix. The position and posture of the end of the robot arm remain unchanged, that is, , by adjusting To optimize the stiffness characteristics of the robot arm. Therefore, formula (29) can be rewritten as: (32) Finally, according to formulas (28) and (32), the stiffness performance index of the robot arm in different configurations can be obtained, and the optimal configuration of the robot arm can be obtained by comparison.

[0005] The adoption of the above scheme can effectively improve the absolute positioning accuracy of the robot arm. By constructing a static stiffness model, calibrating the joint stiffness coefficient, defining the robot arm stiffness characteristic index and combining the zero space stiffness optimization strategy, the optimal group is selected from the countless sets of possible joint configurations of the redundant robot arm for operation, thereby ensuring the stiffness characteristics in the production process and further improving the production quality.

[0006] The beneficial effects of the present invention are: 1. The redundant manipulator positioning accuracy improvement method based on zero space stiffness optimization proposed in the present invention can select the best set of joint configurations from countless sets of redundant manipulators for production operations to improve the production quality of the operations. This method has broad application prospects in the fields of precision manufacturing and the like; BRIEF DESCRIPTION OF THE DRAWINGS

[0007] Figure 1 This is a flow chart of a method for improving the positioning accuracy of redundant manipulators based on zero-space stiffness optimization; Figure 2 This is a simulation diagram of a method for improving the positioning accuracy of redundant manipulators based on zero-space stiffness optimization; Figure 3 It is a bar graph comparing the calibration result with the set value; Figure 4 It is a schematic diagram of the deformed ellipsoid at the end of the robot arm reaching a certain position in space with different configurations; Figure 5 yes The configuration and matrix of the other six joints corresponding to different angles The characteristic value of DETAILED DESCRIPTION

[0008] In order to enable those skilled in the art to better understand the technical solutions in the embodiments of the present invention and to make the advantages of the embodiments of the present invention more obvious and understandable, the technical solutions in the embodiments of the present invention are further described in detail below with reference to the accompanying drawings.

[0009] refer to Figure 2 The following is a simulation diagram of a redundant robot arm positioning accuracy improvement method based on zero space stiffness optimization. In this case, the iiwa seven-degree-of-freedom robot arm is used as an example for simulation verification. The following is the specific operation process and effect. Step 110: Construct a robot arm static stiffness model: According to the parameters of LBR iiwa7 R800 shown in Table 1, α i is the joint torsion angle, a i is the connecting rod length, d i is the connecting rod offset, θ i is the joint rotation angle, M i is the mass of the connecting rod, C i is the center of mass position of the connecting rod; and the joint stiffness coefficients defined in Table 2, the mechanical arm is statically modeled. Step 120 Mechanical arm joint stiffness calibration: In order to excite the various stiffness coefficients of the joints, a spatial force vector is applied to the end of the mechanical arm under different configurations of the mechanical arm. Table 3 shows the torque of each joint and the deformation of the end position of the mechanical arm. The spatial force vector in the table The unit is N, the spatial moment The unit of is N‧mm, and the unit of position is mm. Based on the collected data, the stiffness coefficient of the robot arm is calibrated according to formula (20), and the calibration results of the joint stiffness of the robot arm can be obtained as shown in Table 4. Figure 3 The bar graph is a comparison between the calibration result and the set value. Step 130 defines the stiffness characteristic index of the robot arm: the maximum deformation caused by the unit force vector of the robot arm is the stiffness characteristic index, and the larger the deformation, the lower the stiffness. Step 140 Optimize the stiffness of the redundant robot arm based on the zero space: when the robot arm reaches a certain position in space, adjust the number of redundant degrees of freedom while ensuring that the position of the end of the robot arm remains unchanged. The joint angle value is used to optimize the stiffness. Figure 4 The figure is a schematic diagram of the deformed ellipsoid at the end of the robot arm reaching the position with different configurations. Figure 5 for The configurations of the other six joints at different angles, the matrix The eigenvalue of , after comparison, we can get the position of the robot arm when it reaches this point. The mechanical arm stiffness characteristic corresponding to deg is higher, that is, the corresponding eigenvalue The smallest, so the deformation of the end of the robot arm is smaller and the accuracy is higher when this group of configurations is selected. Table 1. Parameters of the LBR iiwa7 R800 robot arm Table 2. Joint stiffness coefficients defined for the LBR iiwa7 R800 robot arm (N·mm / deg) Table 3. Torque of each joint and end position of the robot arm in different configurations Table 4. Calibration results of joint stiffness of robot arm LBR iiwa7 R800 (N·mm / deg)

Claims

1. A method for improving the positioning accuracy of redundant manipulators based on zero space stiffness optimization, which specifically includes the following four main steps: Step 1: Construct the static stiffness model of the robot arm; Step 2: Calibrate the joint stiffness of the robot arm; Step 3: Define the stiffness characteristic index of the robot arm; Step 4: Optimize the redundant robot arm stiffness based on zero space.

2. According to the method for improving the positioning accuracy of redundant manipulators based on zero-space stiffness optimization according to claim 1, the static stiffness model of the manipulator is constructed as follows: According to the positive kinematics of the robot arm, the end coordinate system of the robot arm can be established Relative to the robot base coordinate system The posture relationship between them is: (1) in is the forward motion equation of the robot arm, is the number of degrees of freedom of the robot arm, Indicates The rotation angle of each joint, Represents the homogeneous transformation matrix of the end of the robot arm relative to the robot arm base. Differentiating formula (1), we can obtain the following relationship: (2) in is the deformation of the robot joint, The position and shape variable vector of the end of the robot arm relative to the base can be converted into a homogeneous transformation matrix: (3) in To convert a vector into a homogeneous transformation matrix function, is the position deformation of the origin of the robot arm end coordinate system relative to the robot arm base coordinate system, is the posture deformation of the robot end coordinate system relative to the robot base coordinate system, where The specific expression is as follows: (4) (5) (6) in , and Respectively represent the coordinate system of the end of the robot arm relative to the coordinate system of the robot arm base and The rotational deformation of the axis. Jacobian matrix ,in Representative The specific expression is: (7) in For the coordinate system { o i x i y i z i }of The direction vector of the axis relative to the machine base coordinate system, Represents the coordinate system { o i x i y i z i }The attitude matrix relative to the base coordinate system, is the position of the end of the robot relative to the coordinate system { o i x i y i z i } location: Due to the weak stiffness characteristics of the robotic arm, the relationship between the deformation of the end of the robotic arm and the deformation of the end load can be determined based on Hooke's law: (8) in is the Cartesian joint stiffness matrix, is the spatial force vector applied to the end of the robotic arm, which is composed of the spatial force and torque At the same time, each joint of the robot arm can be equivalent to a linear spring, as shown in Figure 2. Therefore, the relationship between the joint torque and the joint deformation can be obtained as follows: (9) in is the joint stiffness matrix, is the joint torque. According to the principle of virtual work, the mapping relationship between the end load of the robotic arm and the joint torque of the robotic arm can be constructed as follows: (10) Apply formula (10) to By differentiating, we can obtain the following relationship: (11) Substituting formulas (2), (8) and (9) into formula (11), we can obtain: (12) in To supplement the stiffness matrix, the specific expression is: (13) According to formula (12), the mapping relationship between the Cartesian stiffness matrix and the joint stiffness matrix can be determined as: (14) Usually the supplementary stiffness matrix can be ignored, so formula (14) can be further simplified as: (15) Substituting formula (15) into formula (8), the corresponding static stiffness model can be obtained as follows: (16) 3. After the static stiffness model of the robot arm is constructed according to claim 2, the stiffness coefficient of the robot arm can be calibrated based on the model: In order to avoid solving the inverse of the Jacobian matrix, formula (16) can be rewritten as: (17) For formula (17) By separating the variables, we can obtain the following form: (18) in is the parameter to be calibrated, the matrix The expression is: (19) Finally, the calibration of the joint stiffness of the robot arm can be transformed into an optimization problem to minimize the positioning error of the end of the robot arm: (20) in represents the function to be optimized, r represents the number of measurement points sampled, Indicates j The deformation variables of the end pose of each sampling point are minimized to minimize the deformation variables of the theoretical end pose and the actual measured end pose, so as to achieve the purpose of calibrating the stiffness of the robot joint.

4. After obtaining the joint stiffness of the robot arm according to claim 3, define the robot arm stiffness performance evaluation index according to the joint stiffness: Based on formula (17), we define , it can be proved . For the matrix Blocking can be decomposed into 4 3 3's matrix submodule, namely: (21) in is the displacement stiffness submatrix, is the rotational stiffness submatrix, is the mixed stiffness sub-matrix. Usually the end of the robot arm is only subject to spatial forces The effect of the torque is mainly to produce deformation in the position direction of the end of the robot arm. can be ignored, and the following relationship can be obtained: (22) When the robot is in a certain position When a unit space force is applied to the end of the robot arm , the deformation at the end is related to the direction of the force. According to formula (22), the square of the deformation at the end position can be obtained as: (23) in is the deformation of the end position of the robot under unit spatial force, and the spatial force The following relations are satisfied: (24) In order to solve the maximum deformation caused by unit force, the Lagrange equation is constructed. The specific expression is: (25) in is the Lagrange equation, is the Lagrange multiplier, and the partial derivative of formula (25) is: (26) According to formula (26), For the matrix Substituting formula (26) into formula (23), we can obtain the following relationship: (27) Defining the Matrix The eigenvalue of ,Right now ， , where the eigenvalue and The corresponding eigenvectors are , and . Therefore, the following relationship can be obtained: (28) According to formulas (27) and (28), it can be shown that a unit force is applied to the end of the robot arm along and The directions produce the maximum and minimum deformations respectively, which also means that the greater the deformation, the lower the stiffness in that direction. Then, when the robot arm is in a certain position, the matrix The corresponding maximum eigenvalue is the stiffness performance index of the robot arm.

5. After defining the mechanical arm stiffness characteristic index according to claim 4, the redundant mechanical arm stiffness optimization based on the null space is performed as follows: This patent optimizes the stiffness of redundant robotic arms. The inverse kinematics of redundant robotic arms usually corresponds to an infinite number of sets of solutions, but there is an optimal set of stiffness configurations among these infinite configurations. The patent of this invention adopts the method of zero space optimization to obtain the optimal stiffness configuration of the robotic arm. By introducing the concept of zero space, the motion space of the robotic arm is decomposed into two parts: task space and zero space. In the task space, the robotic arm completes the main task according to the predetermined trajectory and speed; in the zero space, the redundant degrees of freedom are used to optimize the internal characteristics of the robotic arm, thereby improving the overall positioning accuracy of the robotic arm. For formula (2), the corresponding general solution expression is: (29) in represents the generalized joint velocity, is the Jacobian matrix The generalized inverse of is solved as follows: (30) is the null space projection matrix, which is expressed as follows: (31) in is the unit matrix. The position and posture of the end of the robot arm remain unchanged, that is, , by adjusting To optimize the stiffness characteristics of the robot arm. Therefore, formula (29) can be rewritten as: (32) Finally, according to formulas (28) and (32), the stiffness performance index of the manipulator in different configurations can be obtained, and the optimal configuration of the manipulator can be obtained by comparison: The adoption of the above scheme can effectively improve the absolute positioning accuracy of the robot arm. By constructing a static stiffness model, calibrating the joint stiffness coefficient, defining the robot arm stiffness characteristic index and combining the zero space stiffness optimization strategy, the optimal group is selected from the countless sets of possible joint configurations of the redundant robot arm for operation, thereby ensuring the stiffness characteristics in the production process and further improving the production quality.

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