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The robot kinematic model is the basis of motion control, calibration, error analysis, etc. Considering these factors, the kinematic model needs to meet the requirements of completeness, model continuity, and minimality. DH model as the most widely used method to build robot kinematic model still has problems in completeness, model continuity, and calculation, especially for robots with complex mechanisms such as closed chain mechanism and branch mechanism. In this paper, an improved kinematic modeling method is proposed based on the cooperation of the DH model and the Hayati and Mirmirani model and considering the Lie group concept. The improved model is complete and continuous, and when combining with Lie group to calculate, it avoids numbers of trigonometric functions and antitrigonometric functions in the process so as to optimize the algorithm. With this method, the kinematic model of the closed chain cascade manipulator developed in our laboratory is established, and a working process of it is numerically calculated. The results of the numerical calculation are basically consistent with those of virtual prototype simulation, which means the established kinematic model is correct and the numerical calculation method can solve the problem correctly. The kinematic model and the results of the kinematic analysis provide a theoretical basis for the subsequent motion control, calibration, and error analysis of the robot.

Industrial robots are usually composed of the base, the end-effector, and several links connected by joints. The pose of the end-effector of the robot is determined by the angle or displacement of each joint. Robot kinematic model is to establish the relationship between the pose of the end-effector and the angles or displacements of joints which includes the position and velocity equations. The kinematic model is an important factor affecting the accuracy of robot motion, and it is also the basis of the follow-up calibration, error analysis, robot control [

In practical application, due to manufacturing error, assembly error, and other factors, there is often a gap between the actual kinematic parameters of the robot and the design kinematic parameters. So, in many cases, it is necessary to conduct calibration and error analysis on the robot’s kinematic parameters. Considering these factors, a qualified kinematic model is required to meet the requirements of completeness, model continuity, and minimality [

There are many methods to build a robot kinematic model such as vector method, quaternion method, exponential function method, homogeneous matrix method, and so on. Among them, the vector method is the most basic, but the modeling process is tedious and not general which is often used for simple or basic mechanism modeling. The quaternion method uses redundant four parameters to represent the attitude which can describe the attitude changing process with continuous parameters, reduce the number of equations for each target position, and overcome the singularity problem. However, due to its abstraction and other factors, it has not been widely used. The exponential function method is based on the screw theory. In the process of modeling, only the coordinate systems on the base and the target object need to be established, and the description of the component pose is relatively simple. It has become an effective alternative to the traditional vector method and the DH method. But it cannot be directly used when modeling the mobile base robot [

Denavit and Hartenberg proposed the DH method in 1955 [

When the adjacent joint axes are parallel or nearly parallel, the slight change of joint axis direction will lead to the great change of the pose of the common normal line between adjacent axes. Thus, it will result in the discontinuity of DH parameters.

The origin of the link coordinate system cannot be placed at the required position and may be far away from the link. This will cause difficulties in obtaining DH parameters and limitations in error analysis [

DH method is based on a homogeneous transformation matrix, and its possible singular points of trigonometric function will lead to no solution of the kinematic model. Especially, when establishing a kinematic model of the robots including complex mechanisms such as closed chain and branch mechanisms and very complex expression of trigonometric functions and antitrigonometric functions will arise. This makes the calculation extremely difficult.

Aiming at the limitations of the DH method, many scholars have made improvements to it.

On the issue that the kinematic model is of discontinuity when modeling the parallel adjacent joints with the DH method, the common method is to add the rotation transformation to describe the small angle change of joint pose. Hayati and Mirmirani [

On the issue that the position of coordinate system origin is limited when modeling with the DH method, increasing the number of parameters in the model is a common method. Hsu and Everett [

As for solving the problem of calculating the kinematic model established by the DH method, the Lie group theory is adopted in this paper. In recent years, with the development of computational geometry mechanics, the Lie group theory is more and more applied to the kinematic analysis and dynamic analysis of the space rigid body. It is found that when the concept of Lie group and Lie algebra is introduced into the kinematic model based on the homogeneous transformation matrix, the rotation part of the homogeneous transformation matrix is regarded as the three-dimensional rotating group of the Lie group, and the angular velocity corresponding to the rotation part is regarded as the Lie algebra. So, the original homogeneous matrix can be replaced by an orthogonal matrix and the vectors to avoid large numbers of trigonometric functions and antitrigonometric functions in the calculating process. There is an exponential mapping relationship between the Lie group and the rotation angle, and a one-to-one mapping relationship between the Lie group and the Lie algebra. It will greatly reduce the difficulty of kinematic model calculation, especially the difficulty of the velocity solution. Lee [

In this paper, a new kinematic modeling method is proposed based on the cooperation of the DH model and the Hayati and Mirmirani model and considering the Lie group concept. Because increasing the number of model parameters can solve the problem of limitation of coordinate system origin position, this paper changes two translation transformations in the Craig’s DH model to three while the rotation transformation is unchanged. Then, combining the concept of a Lie group, the transformation order in the model is changed to ease the follow-up calculation. The improved model is still discontinuous when the adjacent joints are parallel. So, in this paper, Hayati and Mirmirani model with an added translation parameter is used to model parallel joints. It can be seen from the analysis that the kinematic model proposed in this paper is complete and model continuity and contains 5 independent parameters, which is the least compared with the model in the same condition. With this method, the kinematic model of the closed chain cascade manipulator developed in our laboratory is established, and a working process of it is numerically calculated with the Lie group theory. The results of the numerical calculation are basically consistent with those of virtual prototype simulation, which means the established kinematic model is correct, and the numerical calculation method can solve the problem correctly. The kinematic model and the results of the kinematic analysis provide a theoretical basis for the subsequent motion control, calibration, and error analysis of the robot.

This paper is organized as follows. In Section

The rotation transformation of rigid body in Euclidean space is represented by the rotation matrix

The translation transformation of a rigid body in Euclidean space is represented by the translation matrix

The pose transformation of a rigid body in Euclidean space is represented by the homogeneous transformation matrix

The cross product is a linear operator. For two vectors

There is

The symbol

There is

The DH method only contains four parameters. The origin of the joint coordinate system is determined by the intersection of the common normal line and joint axis, so the origin position of the joint coordinate system cannot be changed. To determine the position and pose of the coordinate system, five parameters are needed. Generally, three translation parameters are used to determine the position of the coordinate system and two rotation parameters are used to determine the attitude of the coordinate system. In this paper, a translation parameter is added to the DH model proposed by Craig [

The modeling method when adjacent joints are not parallel consists of the following steps. The notation of joints, links, and joint coordinate systems are shown in Figure

Number links and joints. Numbering the links sequentially, commencing with the base, identified as 0, and terminating at the end-effector, marked as

Define the joint reference coordinate system. Place the origin of the coordinate system at the desired position on the joint axis. In the Cartesian coordinate system, the

Define reference coordinate systems of the base and the end-effector. According to the actual situation, it can be defined freely on the premise of meeting the required needs and convenient calculation. When transforming from the base coordinate system to link 1, take the method of translating along _{0}, _{0}, and _{0} direction followed by the Euler angle.

Transformation steps between adjacent joint coordinate systems.

Conduct translations on the coordinate system _{i−1}, _{i−1,} and _{i−1} by the distance of a,

Rotate the coordinate system 1 around the axis _{1} by angle _{1} coincide with the joint axis. Thus, the coordinate system 2 is obtained.

Rotate the coordinate system 2 around the axis _{2} by angle _{1} coincide with the axis _{i}. Thus, the coordinate system

So far, the transformation from the coordinate system

The homogeneous transformation matrix from the coordinate system

The coordinate frame assigned by the proposed modeling method when adjacent joints are not parallel.

According to [

Analyzing the model continuity from the geometry aspect is done as shown in Figure _{i−1} and _{i} axis intersect some distance away from the origin of the frame

The joint

To make up for the model discontinuity in Section

The coordinate frame assigned by the modeling method when adjacent joints are parallel.

This modeling method is different from the method in Section

The transformation method between coordinate systems of adjacent joints is as follows.

Translate the coordinate system _{i−1} by the distance to align _{i−1} with _{i}. Coordinate system 1 is obtained.

Coordinate system 2 is obtained from rotating coordinate system 1 around _{1} for angle _{2} for angle _{2} axis with that of joint

Translate the coordinate system 3 along _{3} for the distance

Rotate coordinate system 4 around _{4} for angle _{4} with _{i}. Coordinate system

So far, the transformation from the coordinate system

The homogeneous transformation matrix from the coordinate system

It can be seen from the above that the proposed kinematic modeling method obtained by cooperating Sections

This model not only solves the problem of incompleteness and discontinuity in the establishment of kinematic model with DH method but also introduces the concept of Lie group to avoid the trigonometric function form in the model, which lays a foundation for the subsequent numerical calculation and optimizes the algorithm.

The robot studied in this paper consists of five mechanical arms, and its main structure is shown in Figure

The main structure model of the manipulator.

The main kinematic chain of the manipulator consists of the base and five mechanical arms. The first mechanical arm rotates in the vertical direction relative to the base, so the first mechanical arm is connected to the base by a rotating pair with vertical axis. The five mechanical arms are linked by pin shafts which are treated as rotating pairs. The third and the fourth mechanical arms are all telescopic devices. Each arm has two telescopic arms inside, which can be simplified as one mechanical arm and one telescopic arm are connected by a moving pair.

According to the proposed kinematic modeling method proposed in Section _{0}_{0}_{0}. According to the theory in Section

The coordinate distribution of different joints on the manipulator.

The pose matrix of the coordinate system _{1}_{1}_{1}_{1} relative to the inertial coordinate system _{0}_{0}_{0}_{0} can be expressed as follows:

Then, the attitude and position matrixes of the first mechanical arm in the inertial coordinate system are respectively expressed as

The pose matrix of the coordinate system _{2}_{2}_{2}_{2} relative to coordinate system _{1}_{1}_{1}_{1} can be expressed as follows:

The pose matrix of the coordinate system _{2}_{2}_{2}_{2} relative to the inertial coordinate system _{0}_{0}_{0}_{0} can be expressed as follows:

Then, the attitude and position matrixes of the second mechanical arm in the inertial coordinate system are respectively expressed as

The pose matrix of the coordinate system _{3}_{3}_{3}_{3} relative to coordinate system _{2}_{2}_{2}_{2} can be expressed as follows:

The pose matrix of the coordinate system _{3}_{3}_{3}_{3} relative to the inertial coordinate system _{0}_{0}_{0}_{0} can be expressed as follows:

Then, the attitude and position matrixes of the third manipulator in the inertial coordinate system are respectively expressed as

According to the Lie group theory, _{2} axis by angle _{3} axis by angle _{2} axis and the _{3} axis have the same direction, so that_{3} axis by angle

The pose matrix of the coordinate system _{4}_{4}_{4}_{4} relative to coordinate system _{3}_{3}_{3}_{3} can be expressed as follows:

The pose matrix of the coordinate system _{4}_{4}_{4}_{4} relative to the inertial coordinate system _{0}_{0}_{0}_{0} can be expressed as follows:

Then, the attitude and position matrixes of the telescopic arm of the third arm in the inertial coordinate system are respectively expressed as

The pose matrix of the coordinate system _{5}_{5}_{5}_{5} relative to coordinate system _{4}_{4}_{4}_{4} can be expressed as follows:

The pose matrix of the coordinate system _{5}_{5}_{5}_{5} relative to the inertial coordinate system _{0}_{0}_{0}_{0} can be expressed as follows:

Then, the attitude and position matrixes of the fourth arm in the inertial coordinate system are respectively expressed as

The pose matrix of the coordinate system _{6}_{6}_{6}_{6} relative to coordinate system _{5}_{5}_{5}_{5} can be expressed as follows:

The pose matrix of the coordinate system _{6}_{6}_{6}_{6} relative to the inertial coordinate system _{0}_{0}_{0}_{0} can be expressed as follows:

Then, the attitude and position matrixes of the telescopic arm of the fourth arm in the inertial coordinate system are respectively expressed as

The pose matrix of the coordinate system _{7}_{7}_{7}_{7} relative to coordinate system _{6}_{6}_{6}_{6} can be expressed as follows:

The pose matrix of the coordinate system _{7}_{7}_{7}_{7} relative to the inertial coordinate system _{0}_{0}_{0}_{0} can be expressed as follows:

Then, the attitude and position matrixes of the fifth arm in the inertial coordinate system are respectively expressed as

The pose matrix of the fifth manipulator in the inertial coordinate system is used to determine the pose of the grab in space. Let the geometric center of the end face of the fifth manipulator be point

The four hydraulic cylinders of the manipulator are four branch chains of the robot, which form four closed chains with the main chain manipulator respectively. When establishing the kinematic model, the hydraulic cylinder can be regarded as one component, and the length of hydraulic cylinders 1, 2, 3, and 4 can be set as variables _{1}, _{2}, _{3}, and _{4} respectively. Nominally, joint 8, 9, 10, and 11 are parallel. According to the modeling method in Section

The first closed chain is composed of hydraulic cylinder 1, the first and the second mechanical arms. Equation (

According to the homogeneous transformation matrix of coordinate system 2 relative to coordinate system 1, the position vector of point

Bringing (

Differentiating (

According to equations (

The second closed chain is composed of hydraulic cylinder 2, the second and the third mechanical arms. Equation (

According to the homogeneous transformation matrix of coordinate system 3 relative to coordinate system 2, the position vector of point

Bringing (

Differentiating (

According to equation (

The third closed chain is composed of hydraulic cylinder 3, the third and the fourth mechanical arms. Equation (

According to the homogeneous transformation matrix of coordinate system 5 relative to coordinate system 4, the position vector of point

Bringing (

Differentiating (

According to equation (

The fourth closed chain is composed of hydraulic cylinder 4, the fourth and the fifth mechanical arms. Equation (

According to the homogeneous transformation matrix of coordinate system 7 relative to coordinate system 6, the position vector of point

Bringing (

Differentiating (

According to equation (

Taking a working process of the manipulator for analysis, the actual movement of the working process is as follows. The first mechanical arm rotates to make the mechanical arm turn from the initial position to the target position to grab the target object and then return to the initial position. Next, the second, third, fourth, and fifth mechanical arms cooperate with each other to make the object pass through the wellhead with the specified attitude, and finally, put the object into the pipeline in the required position. According to the kinematic model established in Section

The kinematic parameters of the initial position of the manipulator are as follows:

In 0–5 s, the first mechanical arm rotates at a constant speed to make the end of the arm move from the initial position to the position of the target object. When the position of the object is known, that is the position vector of the mark point

In 5 s–10 s, the end-effector grabs and fixes the target object. The manipulator is in a static state during this period, and the kinematic parameters remain unchanged.

In 10 s–15 s, the first mechanical arm rotates at a constant speed to return to its initial position. The rotating angle of the first mechanical arm is

In 15 s–25 s, the first and second mechanical arms do not move, the third, fourth, and fifth mechanical arms rotate at a constant speed so that the mechanical arms are to the preparation position before entering the pipeline at the moment 25 s. In the process of movement

In 25 s–27 s, the mechanical arm is paused and adjusted.

In 27 s–37 s, the first and second mechanical arms do not move. The third, fourth, and fifth mechanical arms rotate to keep the fifth mechanical arm vertical and enter the narrow part of the pipeline along the vertical straight line at a constant speed. In the process of movement

In 37 s–39 s, the mechanical arm is paused and adjusted.

In 39 s–49 s, the first mechanical arm does not move. The second, third, fourth, and fifth mechanical arms rotate so that the fifth mechanical arm remains vertical and continues to enter the pipeline along the vertical straight line at a constant speed and finally reaches the designated position of the target object. In the process of movement

According to the structural dimension of the manipulator, the structural parameters in its kinematic model can be determined as follows:

According to the kinematic model, structural parameters, and motion parameters of the manipulator, the variation curve of the kinematic parameters of each mechanical arm is shown in Figures

The variation curve of the manipulator joint rotation angle changing with time.

The variation curve of the rotation angle velocity of the manipulator joint rotation angle changing with time.

The variation curve of the hydraulic cylinder joint rotation angle changing with time.

The variation curve of the length of the hydraulic cylinder changing with time.

The variation curve of the telescopic velocity of the hydraulic cylinder changing with time.

According to Figure

According to Figure

According to Figure

According to Figure ^{−1}, 13 s^{−1}, 20 s^{−1}, 30 s^{−1,} and 14 s^{−1}, respectively. During each motion state, the joint angular velocity curves are continuous, smooth, and uniform most of the time, which means that the manipulator has good motion characteristics. The curve is not continuous at the time of motion state connection, and a step occurs, resulting in a sudden increase of angular acceleration. The maximum angular acceleration of the manipulator joint rotation is about 15 s^{−2}, which occurs at 39 s on the third manipulator joint.

According to Figure ^{2}, which appears at 39 s on the place of the hydraulic cylinder 2.

Rotation angle in Figure

According to the above results of kinematic analysis, the following conclusions can be drawn.

Each mechanical arm and each hydraulic cylinder have good motion characteristics within each motion state. The curves of the rotation angle, angular velocity, movement displacement, and movement velocity of each joint with time are continuous and smooth, and the manipulator can run smoothly within each movement state.

At the moment when each motion state is connected, the curves of the rotation angle, angular velocity, movement displacement, and movement velocity of each joint change abruptly with time. At these moments, each joint has acceleration, which will lead to the surge of hydraulic cylinder flow, mechanism vibration and impact, and even the delay of the control system. However, the impact is reduced in the pause stage. Furthermore, at the moment of state connection, the mechanical arm does not require a high motion characteristic.

In this paper, the three-dimensional model of the manipulator is established and imported into Adams for virtual prototype simulation. The results of ADAMS simulation are compared with Matlab calculation results. It is found that the two results are basically the same, which verifies the correctness of the kinematic model and algorithm established in this paper.

The kinematic model of the robot can support the following calibration and error analysis research. The results of the kinematic analysis provide the theoretical basis for the robot’s motion control.

In this paper, a complete and continuous kinematic modeling method is proposed by combining the advantages of existing kinematic modeling methods and introducing the idea of the Lie group. The method is then used to analyze the closed chain cascade manipulator developed in our laboratory and the simulation results of MATLAB and Adams are compared. The conclusions of this paper are as follows:

In this paper, a new kinematic modeling method is proposed based on the cooperation of the DH model and the Hayati and Mirmirani model and considering the Lie group concept. The improved model is complete and continuous and can support the following robot calibration, error analysis, etc.

When the proposed kinematic modeling method is combined with the Lie group theory to conduct numerical calculation, it can avoid numbers of trigonometric functions and antitrigonometric functions in the process. Especially when modeling the robots with complex mechanisms, the complex expression of trigonometric functions and inverse trigonometric functions can make the solution extremely difficult. The singularity of the trigonometric function in the calculation process can lead to no solution. So this strategy can optimize the algorithm.

The proposed kinematic modeling method is used to establish the kinematic model of the closed chain cascaded manipulator developed in our laboratory, and the numerical calculation with the Lie group of its working process is carried out. The results of the numerical calculation are basically consistent with those of virtual prototype simulation in Adams, which means that the established kinematic model is correct and the numerical calculation method can solve the problem correctly. The kinematic model and the results of the kinematic analysis provide a theoretical basis for the robot’s subsequent research, such as motion control, calibration, and error analysis.

As for robots with complex mechanisms and accuracy requirements needed to consider error, the method proposed in this paper can establish the robot kinematic model effectively and support the following calibration and error analysis.

This kinematic modeling method can support the following research of calibration and error analysis. The kinematic model is the theoretical basis for the robot’s motion control, calibration, and error analysis. The corresponding calibration algorithm, error model, and motion control algorithm need to be proposed next. It is very effective in research robots with complex mechanisms and accuracy requirements needed to consider an error.

The data used to support the findings of this study are available from the corresponding author upon request.

The authors declare that there are no conflicts of interest regarding the publication of this article.

This work was supported by the Key Program of Natural Science Foundation of Liaoning Province of China (20170520019).