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

Authors: Yong-Bin Li; Tie-Jun Li; Hui-Fang Zhu; Dong Yang; Ya-Jun Chen; Wen-Ming Zhang; Jian-Ming Liu; Tian Zhang
Journal: Mathematical Problems in Engineering
 (2021)

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## Abstract

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.


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## Body

## 1. Introduction

The main methods of kinematics research on tandem robots today are the DH (Denavit–Hartenberg) parameter method and the exponential formula (POE) of screw theory. The 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]. The 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. They 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. They 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. The 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 pole-climbing 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. They 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. They 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. They designed a master-slave control algorithm and conducted experiments. The 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. They 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? This 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.


## 2. 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-of-freedom hydraulically driven dual-arm robot. It simulates the joints of human arms. The 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. Therefore, 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. The 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 Figure1. 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.Figure 1
Modeling diagram of the humanoid 7-DOF hydraulically driven dual-arm robot.Figure 2
Remote operation control diagram.


## 3. Motion Analysis Based on the DH Parameter Method

### 3.1. DH Parameter Method Modeling and Calculation

The heavy-loading humanoid 7-DOF hydraulically driven dual-arm robot is symmetrically arranged for its arms. Therefore, 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 systemX0Y0Z0. The coordinate system on the n joints is XnYnZn. The origin of coordinate of link n is located at the intersection of the joint n-axis and joint n + 1 axis. The Z-axis of link n coincides with the axis of joint n + 1. The X-axis of link n is at the common normal of the joint n-axis and the joint n + 1 axis. The X-axis is directed from joint n-axis to the joint n + 1 axis. The 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 (Figure3). The 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. an is the length of the link between the joints. θn is the rotation angle of each joint.Figure 3
DH robot arm coordinate system.Table 1
DH robot arm coordinate system parameters.
LinkVariablesdnanαn1θ1a1090°2θ20090°3θ3a2 + a30−90°4θ400−90°5θ5a4 + a5090°6θ60090°7θ70a60According to the introduction to robotics [21], the parameters in Table 1 are brought into formula (1) to obtain the joint transformation matrix An. In formula (1), Rot (Zn−1, θn) is the angle of Xn−1 rotating around the Zn−1-axis by the angle θn. And Trans (0, 0, dn) is the translation of dn along the Zn−1-axis. Trans (an, 0, 0) is the translation of an along the Xn-axis, and Rot (Xn, αn) is the rotation of the αn angle around the Xn-axis.(1)An=RotZn−1,θn1Trans0,0,dn2Transan,0,03RotXn,αn4.According to formula (1),(2)A1=cosθ10sinθ10sinθ10−cosθ10010a10001,A2=cosθ20sinθ20sinθ20−cosθ2001000001,A3=cosθ30−sinθ30sinθ30cosθ300−10a2+a30001,A4=cosθ40−sinθ40sinθ40cosθ400−1000001,A5=cosθ50sinθ50sinθ50−cosθ50010a4+a50001,A6=cosθ60sinθ60sinθ60−cosθ6001000001,A7=cosθ7−sinθ70a6cosθ7sinθ7cosθ70a6sinθ700100001.The formula is gained:(3)T7=A1A2A3A4A5A6A7=r11r12r13q1r21r22r23q2r31r32r33q30001.For details, see Appendix A.


### 3.2. 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]. The mechanical arm structure parameters in Table 2 are substituted into the positive kinematics solution equation (2). The 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.Table 2
DH robot arm parameters.
LinkLength (mm)Joint variablesRange ( )a1940θ1−75∘∼+75∘a2490θ2+60∘∼+180∘a3510θ3−20∘∼+20∘a4240θ4−135∘∼−50∘a5390θ5−40∘∼+40∘a6310θ6+30∘∼+120∘θ7−30∘∼+30∘Note: in order to reflect the generality of the study, the rotation range of each joint in Table2 is smaller than the actual rotation range.Figure 4
DH workspace cloud illustration.
(a)(b)(c)(d)


## 3.1. DH Parameter Method Modeling and Calculation

The heavy-loading humanoid 7-DOF hydraulically driven dual-arm robot is symmetrically arranged for its arms. Therefore, 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 systemX0Y0Z0. The coordinate system on the n joints is XnYnZn. The origin of coordinate of link n is located at the intersection of the joint n-axis and joint n + 1 axis. The Z-axis of link n coincides with the axis of joint n + 1. The X-axis of link n is at the common normal of the joint n-axis and the joint n + 1 axis. The X-axis is directed from joint n-axis to the joint n + 1 axis. The 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 (Figure3). The 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. an is the length of the link between the joints. θn is the rotation angle of each joint.Figure 3
DH robot arm coordinate system.Table 1
DH robot arm coordinate system parameters.
LinkVariablesdnanαn1θ1a1090°2θ20090°3θ3a2 + a30−90°4θ400−90°5θ5a4 + a5090°6θ60090°7θ70a60According to the introduction to robotics [21], the parameters in Table 1 are brought into formula (1) to obtain the joint transformation matrix An. In formula (1), Rot (Zn−1, θn) is the angle of Xn−1 rotating around the Zn−1-axis by the angle θn. And Trans (0, 0, dn) is the translation of dn along the Zn−1-axis. Trans (an, 0, 0) is the translation of an along the Xn-axis, and Rot (Xn, αn) is the rotation of the αn angle around the Xn-axis.(1)An=RotZn−1,θn1Trans0,0,dn2Transan,0,03RotXn,αn4.According to formula (1),(2)A1=cosθ10sinθ10sinθ10−cosθ10010a10001,A2=cosθ20sinθ20sinθ20−cosθ2001000001,A3=cosθ30−sinθ30sinθ30cosθ300−10a2+a30001,A4=cosθ40−sinθ40sinθ40cosθ400−1000001,A5=cosθ50sinθ50sinθ50−cosθ50010a4+a50001,A6=cosθ60sinθ60sinθ60−cosθ6001000001,A7=cosθ7−sinθ70a6cosθ7sinθ7cosθ70a6sinθ700100001.The formula is gained:(3)T7=A1A2A3A4A5A6A7=r11r12r13q1r21r22r23q2r31r32r33q30001.For details, see Appendix A.


## 3.2. 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]. The mechanical arm structure parameters in Table 2 are substituted into the positive kinematics solution equation (2). The 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.Table 2
DH robot arm parameters.
LinkLength (mm)Joint variablesRange ( )a1940θ1−75∘∼+75∘a2490θ2+60∘∼+180∘a3510θ3−20∘∼+20∘a4240θ4−135∘∼−50∘a5390θ5−40∘∼+40∘a6310θ6+30∘∼+120∘θ7−30∘∼+30∘Note: in order to reflect the generality of the study, the rotation range of each joint in Table2 is smaller than the actual rotation range.Figure 4
DH workspace cloud illustration.
(a)(b)(c)(d)


## 4. POE Kinematics Analysis Based on Screw Theory

### 4.1. The Basis of Screw Theory

If the base coordinate system of the rigid body isA 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(4)SE3=RP01:R∈SO3,P∈R3.Here,SE3 is the position and orientation transformation matrix of the B coordinate system with respect to the A coordinate system; R∈SO3 is the orientation transformation matrix of the B coordinate system with respect to the A coordinate system; and P∈R3 is the position vector of the B coordinate system with respect to the A coordinate system.When the rigid body rotates around the rotation axisω=ωXωYωZΤ∈R3, as shown in Figure 1, we can know from the Chasles theorem [11] that it can be realized by a composite motion. The motion is that rotating around the axis ω=ωXωYωZΤ∈R3 for an angle θ and moving for a distance v linearly parallel to the axis ω. The position and orientation transformation is expressed by the matrix index as(5)g=eξ^θ∈SE3.In the formula,ξ^=ω^ν00∈se3,ω,ν∈R3 represents the motion screw; ω^=0−ωZωYωZ0−ωX−ωYωX0is a Lie algebra of SO3, which is called antialgorithm matrix; ν=−ω×q is the unit vector in the direction of movement; q∈R3 is a random point on the rotation axis; se3 is a Lie algebra of SE3; eξ^θ is the matrix index corresponding to ξ^θ. The matrix index is often calculated by using the Rodriguez formula [23].(6)eω^θ=I+ω^sinθ+ω^1−cosθ.where I is the unit matrix of 3×3.According to formula (6), formula (7) can be expanded into the following form:(7)g=eξ^θ=eω^θI+eω^θω×ν+ωωΤνθ01,ω≠0,Iνθ01,ω=0.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 systemB relative to the base coordinate system A is converted to(8)gabθ=eξ^θgab0.where gab0 is the rigid body’ position and orientation change of B and A in the initial pose. gabθ is the final position and orientation change of B relative to A.


### 4.2. POE Modeling and Operation of Screw Theory

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. The end of the movement of the arm is the tool coordinate system. Therefore, the reference configuration of the manipulator in its natural posture is shown in Figure5.Figure 5
Rotational motion of a rigid body.It can be known from Figure6 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(9)gst0=100a2+a3010a1+a4+a5+a600100001.Figure 6
Screw robot arm link reference configuration.From Figure5, establish the spin motion coordinates of each joint.(10)ω1=010,ω2=001,ω3=100,ω4=001,ω5=010,ω6=100,ω7=001.According to the reference configuration of the link in Figure6, the points of each joint axis can be obtained:(11)q1=000,q2=0a11,q3=a2a10,q4=a2+a3a11,q5=a2+a3a1+a40,q6=a2+a3a1+a4+a50,q7=a2+a3a1+a4+a50.Fromξi=ωiqi×ωi [24], we get the unit spin motion of each joint:(12)ξ1=010000Τ,ξ2=010a100Τ,ξ3=01000−a1Τ,ξ4=010a1−a2+a30Τ,ξ5=01000a2+a3Τ,ξ6=01000−a1+a4+a5Τ,ξ6=010a1+a4+a5−a2+a30Τ.From the exponential product formula (7) [24], we can derive(13)eξ^1θ1=cosθ10sinθ100100−sinθ10cosθ100001,eξ^2θ2=cosθ2−sinθ20a1sinθ2sinθ2cosθ20a11−cosθ200100001,eξ^3θ3=10000cosθ3−sinθ3a11−cosθ30sinθ3cosθ3−a1sinθ30001,eξ^4θ4=cosθ4−sinθ40a2+a31−cosθ4+a1sinθ4sinθ4cosθ40a11−cosθ4−a2+a3sinθ400100001,eξ^5θ5=cosθ50sinθ5a2+a31−cosθ50100−sinθ50cosθ5a2+a3sinθ50001,eξ6∧θ6=10000cosθ6−sinθ6a1+a4+a51−cosθ60sinθ6cosθ6−a1+a4+a5sinθ60001,eξ^7θ7=cosθ7−sinθ70a2+a31−cosθ7+a1+a4+a5sinθ7sinθ7cosθ70a1+a4+a51−cosθ7−a2+a3sinθ700100001.Then, the positive solution of the entire kinematics is(14)gSTθ=eξ^1θ1eξ^2θ2eξ^3θ3eξ^4θ4eξ^5θ5eξ^6θ6eξ^7θ7gST0=RθPθ01.IfRθ=r11r12r13r21r22r23r31r32r33,Pθ=P1P2P3. For details, see Appendix B.


### 4.3. POE Kinematics Simulation of Spin Theory

Substitute the structural parameters of the mechanical arm in Table3 into the positive kinematics solution equation (14), which is calculated by the POE of the mechanical arm rotation theory above. Then, 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 two-dimensional point cloud of the range of motion in the Y and Z directions.Table 3
Screw robot arm parameters.
LinkLength (mm)Joint variablesParameter ( )a1940θ1−75∘∼+75∘a2490θ2−30∘∼+90∘a3510θ3−20∘∼+20∘a4240θ4+40∘∼+120∘a5390θ5−40∘∼+40∘a6310θ6−30∘∼+30∘θ7−30∘∼+30∘Figure 7
Screw workspace cloud diagram.
(a)(b)(c)(d)


## 4.1. The Basis of Screw Theory

If the base coordinate system of the rigid body isA 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(4)SE3=RP01:R∈SO3,P∈R3.Here,SE3 is the position and orientation transformation matrix of the B coordinate system with respect to the A coordinate system; R∈SO3 is the orientation transformation matrix of the B coordinate system with respect to the A coordinate system; and P∈R3 is the position vector of the B coordinate system with respect to the A coordinate system.When the rigid body rotates around the rotation axisω=ωXωYωZΤ∈R3, as shown in Figure 1, we can know from the Chasles theorem [11] that it can be realized by a composite motion. The motion is that rotating around the axis ω=ωXωYωZΤ∈R3 for an angle θ and moving for a distance v linearly parallel to the axis ω. The position and orientation transformation is expressed by the matrix index as(5)g=eξ^θ∈SE3.In the formula,ξ^=ω^ν00∈se3,ω,ν∈R3 represents the motion screw; ω^=0−ωZωYωZ0−ωX−ωYωX0is a Lie algebra of SO3, which is called antialgorithm matrix; ν=−ω×q is the unit vector in the direction of movement; q∈R3 is a random point on the rotation axis; se3 is a Lie algebra of SE3; eξ^θ is the matrix index corresponding to ξ^θ. The matrix index is often calculated by using the Rodriguez formula [23].(6)eω^θ=I+ω^sinθ+ω^1−cosθ.where I is the unit matrix of 3×3.According to formula (6), formula (7) can be expanded into the following form:(7)g=eξ^θ=eω^θI+eω^θω×ν+ωωΤνθ01,ω≠0,Iνθ01,ω=0.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 systemB relative to the base coordinate system A is converted to(8)gabθ=eξ^θgab0.where gab0 is the rigid body’ position and orientation change of B and A in the initial pose. gabθ is the final position and orientation change of B relative to A.


## 4.2. POE Modeling and Operation of Screw Theory

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. The end of the movement of the arm is the tool coordinate system. Therefore, the reference configuration of the manipulator in its natural posture is shown in Figure5.Figure 5
Rotational motion of a rigid body.It can be known from Figure6 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(9)gst0=100a2+a3010a1+a4+a5+a600100001.Figure 6
Screw robot arm link reference configuration.From Figure5, establish the spin motion coordinates of each joint.(10)ω1=010,ω2=001,ω3=100,ω4=001,ω5=010,ω6=100,ω7=001.According to the reference configuration of the link in Figure6, the points of each joint axis can be obtained:(11)q1=000,q2=0a11,q3=a2a10,q4=a2+a3a11,q5=a2+a3a1+a40,q6=a2+a3a1+a4+a50,q7=a2+a3a1+a4+a50.Fromξi=ωiqi×ωi [24], we get the unit spin motion of each joint:(12)ξ1=010000Τ,ξ2=010a100Τ,ξ3=01000−a1Τ,ξ4=010a1−a2+a30Τ,ξ5=01000a2+a3Τ,ξ6=01000−a1+a4+a5Τ,ξ6=010a1+a4+a5−a2+a30Τ.From the exponential product formula (7) [24], we can derive(13)eξ^1θ1=cosθ10sinθ100100−sinθ10cosθ100001,eξ^2θ2=cosθ2−sinθ20a1sinθ2sinθ2cosθ20a11−cosθ200100001,eξ^3θ3=10000cosθ3−sinθ3a11−cosθ30sinθ3cosθ3−a1sinθ30001,eξ^4θ4=cosθ4−sinθ40a2+a31−cosθ4+a1sinθ4sinθ4cosθ40a11−cosθ4−a2+a3sinθ400100001,eξ^5θ5=cosθ50sinθ5a2+a31−cosθ50100−sinθ50cosθ5a2+a3sinθ50001,eξ6∧θ6=10000cosθ6−sinθ6a1+a4+a51−cosθ60sinθ6cosθ6−a1+a4+a5sinθ60001,eξ^7θ7=cosθ7−sinθ70a2+a31−cosθ7+a1+a4+a5sinθ7sinθ7cosθ70a1+a4+a51−cosθ7−a2+a3sinθ700100001.Then, the positive solution of the entire kinematics is(14)gSTθ=eξ^1θ1eξ^2θ2eξ^3θ3eξ^4θ4eξ^5θ5eξ^6θ6eξ^7θ7gST0=RθPθ01.IfRθ=r11r12r13r21r22r23r31r32r33,Pθ=P1P2P3. For details, see Appendix B.


## 4.3. POE Kinematics Simulation of Spin Theory

Substitute the structural parameters of the mechanical arm in Table3 into the positive kinematics solution equation (14), which is calculated by the POE of the mechanical arm rotation theory above. Then, 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 two-dimensional point cloud of the range of motion in the Y and Z directions.Table 3
Screw robot arm parameters.
LinkLength (mm)Joint variablesParameter ( )a1940θ1−75∘∼+75∘a2490θ2−30∘∼+90∘a3510θ3−20∘∼+20∘a4240θ4+40∘∼+120∘a5390θ5−40∘∼+40∘a6310θ6−30∘∼+30∘θ7−30∘∼+30∘Figure 7
Screw workspace cloud diagram.
(a)(b)(c)(d)


## 5. Comparative Analysis of Advantages and Disadvantages of the Two Modeling Methods

### 5.1. Advantages and Disadvantages of the DH Parameter Method for Motion Analysis

#### 5.1.1. Advantages

The 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.


#### 5.1.2. Disadvantages

The modeling process is relatively complicated. There are several degrees of freedom to establish several coordinates. The 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. This 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. This is the reason for the inconsistency in the movement ranges of Tables2 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 dn in Table 1 that joints 2 and 3 are at the same area. Therefore, it is easy to produce singularities.


### 5.2. The Pros and Cons of POE Kinematics Analysis with Screw Theory

#### 5.2.1. Advantages

The 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. Then, 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. The 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.


#### 5.2.2. Disadvantages

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


### 5.3. 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. The 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. The DH parameter method has fewer mathematical operations in formula (2) than screw theory in formula (14). The 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).


## 5.1. Advantages and Disadvantages of the DH Parameter Method for Motion Analysis

### 5.1.1. Advantages

The 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.


### 5.1.2. Disadvantages

The modeling process is relatively complicated. There are several degrees of freedom to establish several coordinates. The 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. This 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. This is the reason for the inconsistency in the movement ranges of Tables2 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 dn in Table 1 that joints 2 and 3 are at the same area. Therefore, it is easy to produce singularities.


## 5.1.1. Advantages

The 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.


## 5.1.2. Disadvantages

The modeling process is relatively complicated. There are several degrees of freedom to establish several coordinates. The 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. This 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. This is the reason for the inconsistency in the movement ranges of Tables2 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 dn in Table 1 that joints 2 and 3 are at the same area. Therefore, it is easy to produce singularities.


## 5.2. The Pros and Cons of POE Kinematics Analysis with Screw Theory

### 5.2.1. Advantages

The 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. Then, 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. The 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.


### 5.2.2. Disadvantages

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


## 5.2.1. Advantages

The 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. Then, 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. The 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.


## 5.2.2. Disadvantages

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


## 5.3. 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. The 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. The DH parameter method has fewer mathematical operations in formula (2) than screw theory in formula (14). The 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).


## 6. Conclusion

This 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. Then, 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. The expressions and calculation processes of the two modeling methods are compared. It provides a basis for the modeling method study in the future.



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*Source: 6694621-2021-01-30.xml*