Research on SDH Model Calibration Algorithm for Robotic Arm Based on Differential Transform

With the development of science and technology, it has had a great impact on the manufacturing industry. e manufacturing industry has higher and higher requirements for industrial manipulators, and the expectations for the quality and function of the industrial manipulators in the manufacturing industry are increasing day by day. At the same time, the requirements for the working accuracy of the manipulators are also getting higher and higher. In this paper, a new method is proposed. An error compensation and calibration algorithm based on parameter separation and integration of the standard DHmodel (SDH) is used to construct the kinematic model of the manipulator. e six groups of structural parameters of the six-axis manipulator are calibrated once every two groups, thereby eliminating the inuence of the singular value of the position of the manipulator. Finally, the feasibility of this scheme is veried by simulation experiments, the accuracy is ensured, and the expected purpose is achieved. Because the method in this paper has high requirements on the posture of the manipulator, it reduces the overall calibration calculation and saves time, which has a certain eect on the improvement of the industrial manipulator in the manufacturing industry.


Introduction
Because of the complexity of industrial manipulator components, it is necessary to conduct in-depth research on them so as to realize the e ective operation and control of the manipulator on the basis of mastering the relevant content. e actual accuracy and theoretical design model of the manipulator body may be inaccurate. In order to ensure that the manipulator body conforms to the theoretical design model, improves the positioning and trajectory accuracy of the manipulator, or can completely move according to the program, the manipulator needs to be calibrated after assembly. e accuracy of the kinematic model can be improved if all deviations are re ected in the DH parameters. e main sources of kinematic error in robotic arms include machining error, assembly error, zero-point error, reducer return, reduction ratio error, and calibration error, of which calibration error is limited by the rudimentary calibration device and the streamlined calibration process, and in fact, the largest errors often come from this area. e above calibration methods have two shortcomings: one is that the established robotic arm models are mostly MDH models, which adds an extra parameter to be calibrated, i.e., increases the di culty of calibration. In this paper, an error compensation scheme based on the standard DH model is proposed to address these two problems in the error calibration process.
e traditional standard DH model has the drawback that when two adjacent axes of the arm are parallel or nearly parallel, small errors can cause large changes in the DH structural parameters, resulting in oddities in the arm's position and even damage to the arm. Noting that the di erential transformation matrix of the standard DH model is only related to xed parameters, such as the linkage de ection and linkage rotation angle, and not the joint angle, this greatly simpli es the calculation of the error calibration.
In this paper, a new error calibration and compensation model are developed by making appropriate improvements to the standard DH model, which in turn solves this problem well.

Kinematic Model of a Robotic Arm
During the movement of a robot arm, the angles of the joints of the arm need to be adjusted to achieve the corresponding Cartesian spatial position, and the matrix characterizing this type of transformation relationship is called the mechanical Jacobi matrix. e mechanical Jacobi matrix is only related to the D-H parameters of the arm itself and varies from arm to arm. Suppose a joint is xed and, without any constraints, the joints adjacent to it have six freely changeable dimensions in free space (Cartesian space) with respect to the xed joint, i.e., six degrees of freedom; if an independent coordinate system is established for each joint, then the method of ush coordinate transformation is su cient to solve the problem. However, this is not a good representation of a chain arm system with a unique structure rather than a simple free rigid body, nor is it possible to relate the property parameters of the joints themselves (rotation angle for rotating joints, translation distance for translating joints) to the coordinate transformation.
When expressing the transformation relationship between two coordinate systems, a rotation matrix R and a displacement matrix P are usually required, with a total of 12 parameters. For traditional manipulators, the transformation relationship between coordinate systems can be clearly represented by four parameters θ, d, a, and α, provided that the following two constraints are satis ed. θn represents the axis Xn and axis Xn − 1, dn represents the directional distance between the axis Xn − 1 and axis Xn along the positive direction of the axis Zn − 1, an represents the directional distance between the axis Zn − 1 and axis Zn along the positive direction of the axis X, and α represents axis angle between Zn − 1 and axis Zn. a is the directional distance between the Zn − 1 axis and Zn axis along the Xn axis, and α is the angle between the Zn − 1 axis and Zn axis. e speci c positional relationship and direction of the parameters are shown in Figure 1.
As can be seen in Figure 1, the transformation of the coordinate system O n−1 to the coordinate system O n needs to be done in four steps, rst, rotating the coordinate system O n−1 around Z n−1 by an angle of θ n , translating along the positive direction of Z n−1 by the length of d n , then translating along the positive direction of X n by the length of d n , and nally rotating around X n by an angle of α n Let A n denote the transformation matrix from the coordinate system O n−1 to the coordinate system O n , then A n can be expressed as follows: A n Rot Z n−1, θ n * Trans Z n−1, d n * Trans X n−1, a n * Rot X n−1, α n cθ n −sθ n cα n sθ n sα n a n cθ n sθ n cθ n cα n −cθ n sα n a n sθ n 0 sα n cα n d n where cθ n denotes cosθ n , sθ n denotes sinθ n. ... and so on. e symbols of this type in the equations that indicate the same meaning and are not repeated. e transformation matrix of the 6-axis robot arm is the transformation matrix of each joint multiplied together in order, and by substituting (1), we get (2)

Calibration Algorithm for Errors in the Structural
Parameters of the Robot Arm. e inverse kinematics of a manipulator is the calculation of the angular values of the joints of the manipulator from the known coordinate positions or poses of the end-effectors, also known as the inverse kinematics of the manipulator, which is the inverse of the forward kinematics. e solution to the forward kinematics problem is simple and unique, whereas the solution to the inverse kinematics problem is complex and may have multiple solutions or no solutions at all. Factors that affect the finishing accuracy of a robotic arm include kinematic interpolation, arm load, stiffness, mechanical clearance, tool wear, and thermal effects. An important way to improve the positioning accuracy is to establish a positioning compensation algorithm and an error model. e positioning accuracy of the robotic arm can be divided into two categories: absolute positioning and repeatable positioning accuracy. e absolute positioning accuracy refers to the deviation between the set value and the actual value, and the repeated positioning accuracy refers to the position deviation when the robot arm repeatedly reaches a certain point. Since the repeatable positioning accuracy of robotic arms is generally higher than absolute accuracy, technological breakthroughs in absolute accuracy are particularly important. For this purpose, parameters such as joint stiffness, position error, and temperature deformation of the robot arm need to be identified, an error model or error matrix should be obtained, and the position of the end effector should be servo-corrected. Feedback channels are incorporated into robotic arm servo control, and robotic arm vision servo is one of the methods. e robotic arm visual servo control uses visual feedback sensors to obtain changes in the working environment and image feature extraction through vision processing to obtain the position of the target and the end of the robotic arm in Cartesian space, then the decision control compares the current feedback information with the target information and calculates the corresponding control command to finally guide the robotic arm to complete the task. rough the precision correction algorithm, the 6-axis industrial manipulator system is essentially a semiclosed control structure, which precisely controls the position of the servo motor joint, and the relationship between the motor position and the end effector of the manipulator is determined by kinematics. e relationship between the motor position and the robotic arm end effector position is determined by kinematics. To improve the absolute accuracy of the robotic arm positioning, calibration is necessary because errors inevitably occur between the theoretical kinematics model and the actual model.
In this paper, the posture error model of each joint is constructed and the differential motion model of the overall robot arm is derived by controlling certain joint angles to reduce the effect of singularity.

Differential Errors in Adjacent
Joints. Let Δθi, Δdi, Δai, and Δαi represent the error of joint i, which is the final required result. en, according to the reporter's derivative rule, the differential error transformation matrix between adjacent joints is, easy to get Further, there are In the above equation, Ω i is called the error matrix operator.
In contrast, the MDH model yields an error matrix operator of Comparing (6) and (7), it can be found that the error matrix operator of the SDH model is independent of the joint angle θ i , i.e., when the end position of the arm changes and thus the joints of the arm change, the error matrix operator remains constant, which provides a good condition for the accuracy of the error calibration results and greatly simplifies the operation. e error matrix operator of the MDH model is related to the joint angle θ i , which brings variability to the subsequent calibration algorithm, so this chapter uses the SDH model for error calibration. e recursive equation for the differential error of each joint is derived from equation (6).
It is obvious that expanding the equation dT n to contain multiple dA i cross-multiplying terms can be considered as higher-order infinitesimals with respect to dA i , which can be ignored, and simplifying it further yields dT n � n i�1 A 1 A 2 . . . dA i . . . A n � dT n−1 * A n + T n−1 * dA n � dT n−1 * A n + T n * Ω n (n � 2, 3, 4, 5, 6).

(9)
e equations of the error model are thus established. e parameters Δθi, Δdi, Δai, and Δαi to be calibrated are contained in the matrix Ω n and dT 6 is the direct data obtained from the measuring instruments in the experiment.

Differential Error Model for Inverse Order Joints.
In the previous subsection, a recursive formula for the error in the position of the adjacent joints of the robot arm was obtained. Further processing of equation gives the relationship between the error accumulated at joint i and the error in each joint before joint i.
A closer look at this equation shows that the error at the end of the robot arm dT n * T −1 n is in the form of a sum about the parameters of the individual joints, and that their sums are independent of each other and no coupling terms appear, which provides the preconditions for the elimination method.
It can be seen that T i is the transformation matrix of joint and note that R i Q i R T i is an antisymmetric matrix whose elements are opposite to each other after corresponding along the main diagonal, which greatly reduces the number of equations and thus achieves the effect of reducing the rank of the coefficient matrix.
According to the characteristics of equations (3)-(13), the first n − 2 joints can be controlled so that their joint angles remain constant and only the joint angle n and joint angle n − 1 are changed so that the matrix equations for θn, d n , a n , α n , θ n − １, d n − １, a n − １, and α n − １ can be obtained as follows: Due to the SDH singularity, the coefficient matrices of θ n − １, d n − １, a n − １, and α n − １ are close to singular values, so the calibration results of the structural parameters of joints n − 1 θ n − １, d n − , a n − １, and α n − １ are discarded and only the experimental results of θ n , d n , a n , and α n are retained. In the above equation, [ ]′ and [ ]″ represent the two sets of measurements, respectively, D represents the matrix obtained from the measurements, i.e., the expression on the left side of the equation, and W represents the result obtained by numerical calculation, i.e., the expression on the left side of the equation.
(12) e following is an example of how to solve for the structural parameters θ 6 , d 6 , a 6 , and α 6 for the last joint calibrated, i.e., the 6th joint.
To simplify the calculation, it is assumed that the value of α in the SDH model can only be taken to two common parameter values, 0 and ±π/2. e error matrix equation of (10), which is an equation for the joint angle n and the joint angle n − 1, can be calculated as follows: Mathematical Problems in Engineering  According to the characteristics of the SDH transformation matrix, the rst three equations in equation can be calibrated to Δα 6 and Δa 6 . After the calculation of equation (9), the coe cients before Δθ 5 and Δd 5 are close to zero and can be neglected, and nally a matrix equation is obtained for Δθ 6 , Δd 6 , Δa 6 , Δα 6 , Δa 5 , and Δα 6 , so that the coe cient matrix is not a singular matrix and can be directly calculated to obtain Δθ 6 , Δd 6 , Δa 6 , and Δα 6. e ow chart of the algorithm is shown in Figure 2.
For the error parameters of joints 5, 4, 3, 2, and 1, new measurement data can be collected and calculated by (11) to obtain

Simulation Experiments and Verification Results
e SDH parameters of the robotic arm used in this paper are shown in Table 1.
Experiments were performed using open-loop measurements, where an external measurement device was used to measure the position of an end-mounted actuator or actuator. For each calibration, two measurements are taken for each joint i. A total of 5 × 2 = 10 sets of data are selected to ensure that the joint angles, except joint i and joint i − 1 remain unchanged so that only the di erential transformation information of joint i is preserved after they are substracted. In this loop, the error matrix is calculated and the parameters are calibrated until all parameters are calibrated. In the experimental process, the method in this paper is compared with the ordinary least-square method, which reduces the computational complexity and improves the real-time performance of the system while ensuring calibration accuracy. e data used for comparison are shown in Table 2.
As can be seen from Table 2, in terms of calibration accuracy, the method in this paper is not inferior to the common least-square method, but in terms of the process of calculating the calibration error, the least-square method has to nd the pseudo-inverse of a coe cient matrix of order 4 × 6 to obtain the solution of the chi-square linear equation. In contrast, the method in this paper obtains the nal calibration results by computing ve 6 × 6 full rank matrices after further transformation of the error equation, which greatly simpli es the calculation.
In the course of the experiments, the results of the calibration were compensated for errors with the aid of the inspection data set, and the di erences in position at the end of the robotic arm were obtained by comparing the method of this paper with the least-square method, as shown in  e horizontal coordinates of the above three plots represent the number of the inspection data set, and the vertical coordinates represent the deviation of the end position of the robot arm under the two methods. e calculation method is as follows: the 6 data of each set represent 6 joint angles, which are substituted into (1) and (2) to obtain 6 0 T. e results are then substituted into (5) and (8) to obtain the error di erential matrix after compensation.
Note. e o set angle is the initial rotation angle of the joint and remains constant during the calibration process.

Mathematical Problems in Engineering
From the results in the gure, we can see that the calibration method in this paper is slightly better than the common least-square method, which again veri es the feasibility and e ectiveness of the method in this paper. At the same time, due to the small volume of measurement data collected in this paper, the inverse process of a large matrix is decomposed into the inverse process of several small matrices, which greatly simpli es the calculation and reduces the time cost. Figure 6 shows a comparison of the least-square method and the di erential elimination method proposed in this chapter in terms of calibration time; the di erential elimination method takes almost half the time, and the time spent is more stable.

Conclusion
In this paper, an error calibration optimization method based on the SDH model is proposed by analyzing the composition of the transformation matrix and according to its characteristics.
is method solves the singularity problem of the SDH model in the application of calibration error. It is not only better than the traditional calibration method and least-square method in terms of calibration accuracy but also realizes simple calculation by discarding some abnormal calibration values. Finally, through the veri cation of simulation experiments, it is proved that this method can be used to identify the error of the model parameters of the manipulator and achieve the desired e ect. e source of error is not only the structural parameters of the arm but also many external factors. e calibration accuracy of this method is improved when compared to the least-square method, but it is still in the same order of magnitude and does not signi cantly improve the calibration accuracy, so it cannot be used in demanding working environments. ese aspects will also be further investigated in future research work. Because the method in this paper has high requirements on the posture of the manipulator, it reduces the overall calibration calculation and saves time, which has a certain e ect on the improvement of the industrial manipulator in the manufacturing industry.
Data Availability e data used to support the ndings of this study are available from the corresponding author upon request.

Conflicts of Interest
e authors declare that they have no con icts of interest.      Mathematical Problems in Engineering