Comparative Analysis of the Kinematics Solution Based on the DH Method and Screw Theory

The premise of analyzing and researching robot technology is to establish a proper mathematical model and then to solve it with kinematics. In this study, a self-developed humanoid hydraulically driven dual-arm robot is taken as the research object, and the DH (Denavit–Hartenberg) parameter method and the rotational exponential formula (POE) are used to solve the kinematics of the robot. The calculation results are veriﬁed by simulation. The advantages and disadvantages of the two methods are analyzed. The diﬀerences between the two methods are compared. It lays a foundation for other scholars to choose mathematical models when analyzing the mechanism in the future.


Introduction
e main methods of kinematics research on tandem robots today are the DH (Denavit-Hartenberg) parameter method and the exponential formula (POE) of screw theory. e DH parameter method was proposed by Denavit and Hartenberg of the University of Chicago in the United States in 1955. It was named in honor of the two scientists [1]. e research on the theory of curl entered a heyday in the nineteenth century. Among them, a comprehensive study and discussion of screw theory was first conducted by Ball, professor of the University of Cambridge in the United Kingdom. And a classic work called "screw theory" was completed [2,3].
At present, scholars at home and abroad have performed plenty of research studies on the DH method application in the tandem field. ey have obtained a batch of research results. Jian et al. [4] used the DH method to solve the coordinate transformation of the serial six-degree-of-freedom robot arm. ey established the transformation relationship between the robot arm and image mark. It laid a foundation for the study of accuracy error when the robot arm grabbed the workpiece on a repeated basis. Li et al. [5] used the DH method to establish the motion model of the dual-arm robot. e model was direct and vivid in geometric sense for motion parameters. It laid the foundation for the robot's motion calibration error analysis. Qiaoling et al. [6] used the DH method to establish a kinematic model of the open-chain pole-climbing robot. It provided a theoretical basis for improving the motion accuracy control of the poleclimbing robot. Wen [7] compared the advantages and disadvantages of the standard DH method and the improved one in the process of establishing multilink coordinate systems. It provided a reference basis for engineers to use different forms of DH methods when establishing mathematical models of tandem robots. Yang et al. [8] used the DH method to express the relationship between speed and position of each part of the wave glider. It broadened the scope of DH method application. Zhao et al. [9] took the photographic robots as research objects which are used in the virtual studio. According to the actual needs of the studio or stage, they proposed a new analysis method for the working end of photographic robot through the improved DH method. Kelemen et al. [10,11] introduced a new algorithm for the control of kinematically redundant manipulator considering three secondary tasks.
Similarly, scholars at home and abroad have performed a lot of research studies on the application of screw theory in the kinematics of the series mechanism. Wei et al. [12] proposed a serpentine robot kinematics modeling method based on screw theory. It proved that the screw theory modeling method was efficient. Li et al. [13] analyzed the positive kinematics and operability of the tandem manipulator by using screw theory, and they proved that screw theory was feasible. Liu et al. [14] used screw theory to analyze the motion of the manipulator with four freedoms in series. ey deduced the singular configuration of the manipulator to avoid singular positions. Zhu [15] solved the problem of kinematic analysis of five-degree-of-freedom tandem welding robots by using screw theory. Yang et al. [16] proposed a method for evaluating and predicting the machining accuracy of multiaxis CNC machine tools based on screw theory. ey established the kinematics model of each axis of machine tool and used the distance error formula to derive a model for general machining allowance error. It can greatly simplify the motion error analysis of multiaxis CNC machine tools. Wang et al. [17] used screw theory of the exponential product (POE) formula to solve the positive kinematics problem of the tandem free-floating space robot system. It avoided the introduction of inertia tensor parameters and broadened the application range of the theory. Niu et al. [18] established the positive kinematics model of the minimally invasive abdominal cavity surgery robot by using screw theory. ey designed a master-slave control algorithm and conducted experiments. e maximum spatial position error is less than 1 mm. Qu et al. [19] proposed a unified model of open-chain robot output accuracy based on the Lie group and Lie algebra considering joint clearance and structural parameters. ey used a six-degree-of-freedom open-chain manipulator to verify and analyze the model and to prove the model was effective. Chen and Zhou [20] analyzed the kinematics of a serial modular redundant robot based on screw theory; they simplified the kinematic modeling process of modular robots with arbitrary configurations and degrees of freedom.
In summary, the premise for analyzing and researching robotics is to establish a proper mathematical model. For the DH method and screw theory, which is more concise and specific for modeling its motion characteristics (controlling the mapping of joints to end motions), what are their respective advantages? is study takes the self-developed high-load humanoid 7-DOF hydraulically driven dual-arm robot as the research object and uses the DH parameter method and the exponential formula of spin theory to establish a mathematical model of the robot. In this way, the kinematics model and calculation process are fully compared, and their respective advantages are demonstrated.

Introduction of the Humanoid Hydraulically Driven Dual-Arm Robot
With the funding of the National Key R&D Program, the team independently developed a large-load seven-degree-offreedom hydraulically driven dual-arm robot. It simulates the joints of human arms. e overall structure of the human arm is composed of three joints of the shoulder, elbow, and wrist with seven degrees of freedom. Among them, there are three shoulder joints, three wrist joints, and one elbow joint, respectively. By considering the arrangement and motion range of human arm joints, the hydraulic two-arm robot can be designed in a more flexible way, and it can avoid obstacles in the space. erefore, the robot also has seven axes. Among them, the first axis, the second axis, the fourth axis, the sixth axis, and the seventh axis are made of telescopic hydraulic cylinders. In order to achieve the mobility of human arm joints, the connecting part adopts a four-link mechanism. e third and fifth axes use swing hydraulic cylinders to increase the compactness of the structure. According to the layout and parameters of each joint, a three-dimensional modeling of the dual-arm robot is performed to complete the detailed design of the robot structure, as shown in Figure 1. At the same time, it is also designed to wear a remote operation device with two arms to achieve flexible control of human body intention construction, as shown in Figure 2.

DH Parameter Method Modeling and Calculation.
e heavy-loading humanoid 7-DOF hydraulically driven dual-arm robot is symmetrically arranged for its arms. erefore, only one mechanical arm needs to be analyzed. A robotic arm is equivalent to rods and joints with the same function. According to the DH coordinate system construction rule, the establishment of the robotic arm coordinate system is as follows: the joint of the robot arm to the box is used as the initial base coordinate system X 0 Y 0 Z 0 . e coordinate system on the n joints is X n Y n Z n . e origin of coordinate of link n is located at the intersection of the joint n-axis and joint n + 1 axis. e Z-axis of link n coincides with the axis of joint n + 1. e X-axis of link n is at the common normal of the joint n-axis and the joint n + 1 axis. e X-axis is directed from joint n-axis to the joint n + 1 axis. e Y-axis of the link n is determined by the right-hand rule [21].
According to the provisions of the previous paragraph, the coordinate system is established for each joint of the robot arm. In order to simplify the establishment of the coordinate system, the robotic arm is straightened and modeled as the robotic arm DH coordinate system (Figure 3). e DH parameters of the right drill arm of the top plate are obtained from the design dimensions of the robotic arm, as shown in Table 1. a n is the length of the link between the joints. θ n is the rotation angle of each joint.
According to the introduction to robotics [21], the parameters in Table 1 are brought into formula (1) to obtain the joint transformation matrix A n . In formula (1), Rot (Z n− 1 , θ n ) is the angle of X n− 1 rotating around the Z n− 1 -axis by the angle θ n . And Trans (0, 0, d n ) is the translation of d n along the Z n− 1 -axis. Trans (a n , 0, 0) is the translation of an along the X n -axis, and Rot (X n , α n ) is the rotation of the α n angle around the X n -axis.

DH Parameter Method Motion
Simulation. MATLAB is a powerful commercial mathematical software, which is mainly used in data analysis, numerical calculation, and data visualization. It is currently widely used in scientific research at home and abroad. By using the numerical calculation function and data visualization function in MATLAB, it is possible to efficiently and quickly solve the cloud map of the robot's workspace [22]. e mechanical arm structure parameters in Table 2 are substituted into the positive kinematics solution equation (2). e equation is calculated by the DH parameter method of the robot arm to draw the working range of the robot arm, as shown in Figure 4. In Figure 4, a is the three-dimensional point cloud of the range of motion of the robotic arm, b is the two-dimensional point cloud of the range of motion in the X and Z directions, c is the two-dimensional point cloud of the range of motion in the Z and Y directions, and d is the two-dimensional point cloud of the range of motion in the Z and Y directions.

e Basis of Screw eory.
If the base coordinate system of the rigid body is A and the dynamic coordinate system is B, then the set of position and orientation transformations of the rigid body on A can be expressed as Table 1: DH robot arm coordinate system parameters.

Link
Variables d n a n α n Rotation joint of parallel paper Rotation joint of parallel and vertical paper Rotation joint of vertical paper   Mathematical Problems in Engineering Here, SE (3) is the position and orientation transformation matrix of the B coordinate system with respect to the A coordinate system; R ∈ SO(3) is the orientation transformation matrix of the B coordinate system with respect to the A coordinate system; and P ∈ R 3 is the position vector of the B coordinate system with respect to the A coordinate system.
When the rigid body rotates around the rotation axis ω � ω X ω Y ω Z Τ ∈ R 3 , as shown in Figure 1, we can know from the Chasles theorem [11] that it can be realized by a composite motion. e motion is that rotating around the axis ω � ω X ω Y ω Z Τ ∈ R 3 for an angle θ and moving for a distance v linearly parallel to the axis ω. e position and orientation transformation is expressed by the matrix index as g � e ξθ ∈ SE(3).
In the formula, ξ � ω ] 0 0 ∈ se(3), ω, ] ∈ R 3 represents the motion screw; ω � Note: in order to reflect the generality of the study, the rotation range of each joint in Table 2 is smaller than the actual rotation range. Mathematical Problems in Engineering matrix index is often calculated by using the Rodriguez formula [23].
e ωθ � I + ω sin θ + ω(1 − cos θ). (6) where I is the unit matrix of 3 × 3. According to formula (6), formula (7) can be expanded into the following form: As described above, the index of the motion screw can represent the relative motion of the rigid body. It is that after the screw motion on the rigid body, the position and orientation transformation of the moving coordinate system B relative to the base coordinate system A is converted to g ab (θ) � e ξθ g ab (0). (8) where g ab (0) is the rigid body' position and orientation change of B and A in the initial pose. g ab (θ) is the final position and orientation change of B relative to A.

POE Modeling and Operation of Screw eory.
When establishing the rotation coordinate system of each joint of screw theory, it is necessary to select one part of the robot arm as reference and one part as the end of the movement.
To this end, we set the joint of the robot arm to the box as the inertial coordinate system. e end of the movement of the arm is the tool coordinate system. erefore, the reference configuration of the manipulator in its natural posture is shown in Figure 5. It can be known from Figure 6 that the starting joint of the robot arm is the inertial coordinate system S, and the end is the tool coordinate system T. When it is zero, the initial configuration of each joint is From Figure 5, establish the spin motion coordinates of each joint.

POE Kinematics Simulation of Spin eory.
Substitute the structural parameters of the mechanical arm in Table 3 into the positive kinematics solution equation (14), which is calculated by the POE of the mechanical arm rotation theory above. en, draw the working range of the mechanical arm, as shown in Figure 7. In Figure 7, a is the three-dimensional point cloud of the range of motion of the robotic arm, b is the two-dimensional point cloud of the range of motion in the X and Y directions, c is the two-dimensional point cloud of the range of motion in the X and Z directions, and d is the twodimensional point cloud of the range of motion in the Y and Z directions.

Advantages and Disadvantages of the DH Parameter
Method for Motion Analysis 5.1.1. Advantages. e DH parameter method is simpler than the POE of screw theory, and its calculation process is simple. At the same time, the method is highly versatile and does not require the structural order and complexity of the robot.

Disadvantages.
e modeling process is relatively complicated.
ere are several degrees of freedom to establish several coordinates. e latter coordinate must be able to be transformed to the previous one. In addition, the attitude of the mechanism will also make modeling more difficult. is study chooses the initial posture as a straight line. While seeking the best modeling posture, a problem appears. It is that when the initial posture changes, the range of motion of each joint will change accordingly. is is the reason for the inconsistency in the movement ranges of Tables 2 and 3. One of the biggest disadvantages is that except for the standard Stanford robot, the DH parameter method cannot truly represent the initial coordinate position of each joint of the robot. It can be seen from d n in Table 1 that joints 2 and 3 are at the same area. erefore, it is easy to produce singularities.   Advantages. e POE of screw theory only needs two coordinate systems, namely, the inertial coordinate system and the tool coordinate system. In this way, the motion of the rigid body can be described as a whole in the inertial coordinate system. en, a complete and obvious geometric description can be provided. Compared with the establishment of multiple coordinate systems of the DH parameter method, it is more global and complete. e state of motion also overcomes the singularity produced by the local coordinate system of the DH parameter method; establishing a model in any attitude does not affect the ease of modeling, which greatly simplifies the analysis of complex mechanisms.

Disadvantages.
e POE of screw theory is a bit more difficult to get started than the DH parameter method, and the calculation process is cumbersome.

Comparison of the Expressions and Calculation
Processes of the Two Modeling Methods. Compared with the DH parameter method (the coordinate system diagram), screw theory is more concise, considering its reference configuration diagram and the mathematical symbols; the key of the DH parameter method is the establishment of the coordinate parameter table. e key of screw theory is that the calculation process should not be wrong; the DH parameter method is more concise than screw theory with respect to the calculation symbols and calculation results; as for the motion-space-simulation cloud image, the DH parameter method generates the image according to the direction of the end coordinate system, while screw theory generates on the basis of the initial coordinate direction. e DH parameter method has fewer mathematical operations in formula (2) than screw theory in formula (14). e DH parameter method uses the formula (2) to simulate the three-dimensional space cloud image with the same time as the screw theory formula (14).

Conclusion
is study first introduces the current research status of the DH parameter method and screw theory index formula in the field of series and parallel mechanisms in recent years. en, a large load humanoid 7-DOF hydraulically driven double-arm robot with a large load is introduced. Second, the DH parameter method and screw theory exponential formula are used to model the research object, calculate the solution, and simulate the results. Finally, the advantages and disadvantages of the two methods are analyzed, respectively. e expressions and calculation processes of the two modeling methods are compared. It provides a basis for the modeling method study in the future.

Data Availability
e data used to support the findings of this study are included within the article.