Kinematic Analysis of Six-Axis Industrial Robots: Unlocking Precision in Intelligent Manufacturing

In the era of Industry 4.0 and Made in China 2025, artificial intelligence has become the core driving force of industrial transformation, and industrial robots, as the core carrier of intelligent manufacturing, have reshaped the production mode of traditional manufacturing with their high efficiency, precision and flexibility. Among various industrial robot types, six-axis industrial robots stand out for their six serial joints and multi-degree-of-freedom motion capabilities, becoming an irreplaceable key equipment in fields such as automotive manufacturing, electronic assembly, and precision machining. A in-depth research on the kinematic characteristics of six-axis industrial robots, conducted by Zhu Yikun from the School of Mechanical Engineering and Automation of Wuhan Textile University, has made important progress in clarifying the motion mechanism of such robots, optimizing their motion control strategies and improving their operational precision, providing a solid theoretical basis for the further popularization and application of six-axis industrial robots in intelligent manufacturing.

The research, focusing on the structural and motion characteristics of six-axis industrial robots, starts from the perspective of kinematics and conducts a systematic and detailed analysis of the robot’s motion laws, mathematical modeling, and equation verification. Six-axis industrial robots are characterized by a series structure of six joints, each equipped with an independent driver that controls the joint’s rotation and movement, enabling the robot’s end effector to complete complex spatial motion trajectories. Different from traditional industrial robots with fewer degrees of freedom, six-axis industrial robots have significant structural advantages: their big arm adopts a hollow design, which allows all external equipment and cables to be built into the arm, greatly reducing the overall weight of the robot while enhancing its adaptability to different working sites and improving its motion flexibility. In terms of installation, the six-axis industrial robot supports two flexible installation methods: upright installation for conventional working conditions and inverted installation on the ceiling for special working scenarios, which further expands its application scope in industrial production. Taking the FANUC ARC100i six-axis industrial robot as a typical research case, this robot has a standard 6 degrees of freedom, a repeat positioning accuracy of 0.08 mm, and a load capacity of 6 kg at the arm and wrist, which can well meet the precision operation requirements of most industrial production links and is a representative model in the current six-axis industrial robot market.

To realize the precise control of the six-axis industrial robot’s end effector and make it complete the expected spatial motion to operate parts and tools, the research takes the coordinate system as the core mathematical research carrier, and constructs a complete mathematical theoretical system for the robot’s motion description by defining the position, posture and coordinate mapping relationship of the robot in the space. Position and posture are the basic physical quantities to describe the motion state of the robot’s end effector in the space. In the research, a 3×1 position vector is used to represent any point in a selected Cartesian coordinate system, and three mutually orthogonal unit vectors are adopted to characterize the posture of the end effector in the coordinate system. For a certain point P in the space, its position relative to the Cartesian coordinate system {A} is expressed by the position vector ^A P, which is composed of the three coordinate components of the point in the X, Y and Z axes of the coordinate system, realizing the quantitative description of the point’s spatial position. When a new Cartesian coordinate system {B} is established in the space, the origin of the coordinate system {B} may coincide with or deviate from that of the coordinate system {A}, and the unit vectors of the X, Y and Z axes of the coordinate system {B} relative to the coordinate system {A} form a rotation matrix _B^A R. This rotation matrix is a 3×3 matrix composed of the direction cosines of the coordinate axes of the new system relative to the original system, which is the core mathematical tool for describing the posture transformation between different coordinate systems.

Coordinate mapping is the key link to realize the conversion of the robot’s motion parameters between different coordinate systems, and it is the basis for establishing the kinematic model of the six-axis industrial robot. In the actual motion process of the robot, the coordinate systems established for different joints and links often do not coincide, and there are both translational offset and rotational angle deviation between the origins. Therefore, the coordinate mapping of the six-axis industrial robot needs to combine two basic transformation forms: translation and rotation. The research points out that the conventional coordinate system mapping of the six-axis industrial robot needs to complete the rotation around the X, Y and Z axes first, and then perform the translational transformation along the three axes. The rotation transformation is realized by the rotation matrix Rot(x,α), Rot(y,β) and Rot(z,γ) around the three axes, and the translational transformation is expressed by the translation vector Trans(d1+d2+d3) composed of the translational distances along the three axes. The superposition of the two transformations forms the complete coordinate mapping relationship of the robot’s space motion, which provides a mathematical basis for the subsequent establishment of the kinematic equation of the six-axis industrial robot by unifying the motion parameters of different joints and links into the same global coordinate system.

On the basis of the mathematical theoretical system, the research constructs a detailed kinematic model of the six-axis industrial robot by analyzing the link parameters and using the Denavit-Hartenberg (D-H) parameter method, which is the most widely used modeling method in the kinematic analysis of industrial robots. The six-axis industrial robot is regarded as an open-loop link mechanism with six rotating joints and six links in the research, and the base of the robot is defined as link 0 that does not belong to the moving link structure, and the other moving links are numbered from 1 to 6 in the order of their connection with the base and each other. A fixed base coordinate system is established on the robot’s fixed base, and a standard local coordinate system is built on each of the six moving links. The transformation relationship between adjacent local coordinate systems is described by the homogeneous transformation matrix, and the elements of the homogeneous transformation matrix are determined by four core D-H link parameters: link twist angle, link length, link offset and joint angle. These four parameters respectively characterize the spatial geometric relationship between adjacent links and the motion characteristic parameters of the joints, and are the key to quantifying the motion of each link and joint of the six-axis industrial robot.

In the process of establishing the D-H coordinate system for the six-axis industrial robot, the research follows the strict D-H parameter setting rules: the extension line of the joint axis is drawn first, the common perpendicular line between the axis i and the axis i+1 of the adjacent joints is constructed, and the intersection point of the common perpendicular line and the joint axis i is taken as the origin of the local coordinate system of the link i; the joint axis i is defined as the Z axis of the local coordinate system, the common perpendicular line is the X axis, and the Y axis is determined by the right-hand rule of the Cartesian coordinate system, which ensures the standardization and uniqueness of the established coordinate system and lays a solid foundation for the accurate calculation of the homogeneous transformation matrix. For the six-axis industrial robot, the waist structure is defined as joint 1 with the joint angle θ1, the shoulder structure is joint 2 with the joint angle θ2, and the remaining joints are numbered and their joint angles are defined in turn according to the structural order of the robot, forming a complete joint motion parameter system. The research also collates the initial position parameters of each link of the six-axis industrial robot according to its actual use instructions, including the specific values of link twist angle, link length, link offset and initial joint angle of each link, as well as the variable range of each joint angle. For example, the joint angle range of joint 1 is -180° to 180°, the joint angle range of joint 2 is -90° to 120°, and the joint angle range of joint 6 can reach -360° to 360°, which reflects the different motion ranges and flexibility of each joint of the six-axis industrial robot, and provides accurate parameter basis for the subsequent establishment of the kinematic equation.

Based on the D-H coordinate system and link parameters, the research further constructs the kinematic equation of the six-axis industrial robot by using the homogeneous transformation and chain transformation principle. In order to clearly describe the spatial position relationship between the coordinate systems, three intermediate coordinate systems are defined for each link in the modeling process, and the transformation relationship between the intermediate coordinate systems and the local coordinate systems of the links is determined by rotation and translation transformation along the joint axes. The homogeneous transformation matrix ^i-1T from the local coordinate system of link i-1 to the local coordinate system of link i is the product of four basic transformation matrices: the rotation matrix Rot(z,θi) around the Z axis by the joint angle θi, the translation matrix Trans(0,0,di) along the Z axis by the link offset di, the translation matrix Trans(ai-1,0,0) along the X axis by the link length ai-1, and the rotation matrix Rot(x,αi-1) around the X axis by the link twist angle αi-1. This homogeneous transformation matrix contains all the D-H parameters of the adjacent links and joints, and can accurately describe the spatial motion transformation from the previous link to the next link.

According to the chain transformation principle of the open-loop link mechanism, the overall homogeneous transformation matrix _6^0T from the base coordinate system (link 0) to the end effector coordinate system (link 6) of the six-axis industrial robot is the product of the homogeneous transformation matrices between all adjacent links, that is, _6^0T(θ1,θ2,θ3,θ4,θ5,θ6) = _1^0T(θ1)·_2^1T(θ2)·_3^2T(θ3)·_4^3T(θ4)·_5^4T(θ5)·_6^5T(θ6). This overall homogeneous transformation matrix is a function of the six joint angles of the robot, and its elements contain the complete information of the position and posture of the robot’s end effector in the global coordinate system: the first three rows and three columns of the matrix form the rotation matrix describing the posture of the end effector, and the first three rows and the fourth column form the position vector describing the spatial position of the end effector. The research further deduces the specific expression formulas of the position and posture parameters of the end effector by expanding the overall homogeneous transformation matrix, in which the position parameters Px, Py, Pz are the functions of the link length, link offset and joint angles of the robot, and the posture parameters nx, ny, nz, ox, oy, oz, ax, ay, az are the composite functions of the sine and cosine of the joint angles and their combined angles (such as θ2+θ3). These specific expression formulas realize the complete quantitative mapping from the robot’s joint motion parameters to the spatial motion state of the end effector, and form the core positive kinematic equation of the six-axis industrial robot. When all joint variables of the robot are 0 at the initial position, the research calculates the specific value of the overall homogeneous transformation matrix according to the D-H parameters, which verifies the rationality of the kinematic equation at the initial state of the robot.

To verify the correctness and accuracy of the established kinematic equation of the six-axis industrial robot, the research adopts MATLAB software to carry out kinematic simulation and verification, taking the positive kinematics as the core verification object. MATLAB is a powerful scientific computing and data visualization software, which has significant advantages in processing complex mathematical calculations, building mathematical models and realizing graphical display of data. It can free researchers from tedious manual mathematical calculations, and complete the rapid solution and simulation of the robot’s kinematic equation through its built-in algorithm and toolbox, which is the most commonly used software in the current industrial robot kinematic research. The simulation verification adopts a comparative research method, which is divided into two core steps: first, on the basis of mastering the rotation angles of each joint axis of the six-axis industrial robot in the initial state, the relevant parameters are substituted into the manually deduced positive kinematic equation for manual calculation to obtain the theoretical calculation results of the joint angles and the position and posture parameters of the end effector; second, the same initial parameters are input into the MATLAB simulation program, and the software is used to carry out the automatic solution of the kinematic equation to obtain the simulation calculation results. By comparing the consistency of the manual calculation results and the software simulation results, the correctness of the kinematic equation is verified.

The simulation verification results show that the manual calculation results of the six joint angles of the six-axis industrial robot are basically consistent with the simulation calculation results obtained by MATLAB software, and the error between the two is extremely small and can be ignored, which fully proves that the positive and inverse kinematic equations of the six-axis industrial robot established in the research are correct and effective. For example, the manual calculation result of a certain joint angle is 1.6522 rad, and the MATLAB simulation result is 1.650964 rad; the manual calculation result of another joint angle is -0.8745 rad, and the simulation result is -0.874498 rad. The high consistency of the results not only verifies the rationality of the mathematical modeling and equation deduction process, but also provides a reliable theoretical basis for the actual motion control of the six-axis industrial robot, ensuring that the robot can accurately reach the expected spatial position and posture according to the set joint motion parameters in the actual operation process.

The in-depth kinematic analysis of six-axis industrial robots conducted in this research has important theoretical and practical significance for the development of the industrial robot industry and the promotion of intelligent manufacturing. In terms of theoretical significance, the research perfects the kinematic theoretical system of six-axis industrial robots, clarifies the mathematical relationship between the joint motion parameters and the spatial motion state of the end effector, and provides a standardized modeling and analysis method for the kinematic research of similar multi-axis industrial robots. The research method taking the D-H parameter method as the core and combining homogeneous transformation and chain transformation principle can be extended to the kinematic analysis of industrial robots with more degrees of freedom, which enriches the theoretical research means of industrial robot kinematics. In terms of practical significance, the correct and effective kinematic equation established in the research can directly guide the motion control and trajectory planning of six-axis industrial robots in actual industrial production. By inputting the expected position and posture parameters of the end effector into the kinematic equation, the required joint motion parameters can be accurately solved, which realizes the precise control of the robot’s motion trajectory and improves the operation precision and production efficiency of the robot. At the same time, the research results can also provide a theoretical basis for the structural optimization and performance improvement of six-axis industrial robots. For example, according to the kinematic characteristics of each joint, the structural design of the joint and link can be optimized to further improve the motion flexibility and load capacity of the robot, and expand its application in high-precision and complex industrial operation scenarios.

In the context of the deep integration of Industry 4.0 and Made in China 2025, the demand for industrial robots in the manufacturing industry is increasing day by day, and the six-axis industrial robot, as a high-precision and multi-functional intelligent equipment, will play a more important role in the industrial transformation and upgrading. The kinematic analysis results of six-axis industrial robots provide a solid technical support for the further intelligent upgrading of industrial robots. On the basis of this research, the follow-up research can further combine artificial intelligence, machine learning and other technologies to realize the adaptive trajectory planning and real-time motion control of six-axis industrial robots, make the robot better adapt to the dynamic and complex industrial production environment, and further improve the intelligence level and production efficiency of the manufacturing industry. At the same time, the research results can also promote the popularization and application of six-axis industrial robots in more emerging fields such as aerospace, medical equipment manufacturing and precision instrument processing, and inject new impetus into the high-quality development of China’s manufacturing industry.

In general, the kinematic analysis of six-axis industrial robots conducted by Zhu Yikun has made important breakthroughs in the mathematical modeling and equation verification of such robots, which is an important achievement in the field of industrial robot research. The research results not only have important theoretical guiding significance for the kinematic research of industrial robots, but also have strong practical application value for the actual production and application of six-axis industrial robots, and will play a positive role in promoting the development of China’s intelligent manufacturing industry and the realization of the Made in China 2025 strategy.

Author: Zhu Yikun Affiliation: School of Mechanical Engineering and Automation, Wuhan Textile University, Wuhan 430073, Hubei Province, China Journal: Technology Innovation and Application DOI: 10.19981/j.T2095-2945.2021.27.023 (Word count: 3896)