## 1 Introduction  
Industrial robots play an important role in manufacturing as manufacturing automation and intelligence continuously improve. The industrial robot industry is an important indicator of a country's manufacturing level and technology level. Industrial robots are widely used in different fields, such as automobile manufacturing, electrical and electronics, and aerospace. In addition, industrial robots are core pieces of equipment for the industrial intelligent manufacturing. The workspace of a robot is an important index of the flexibility of the robot. It provides the necessary constraint information for the path planning and control of the robot. It has important significance in the structural design and path planning. Kinematics represent the relationship between the position and the pose of the end-effector and the joint variables. This lays the basis for the workspace of the multijoint industrial robots. The Denavit-Hartenberg (D-H) parametric method is a standard method of modeling [1; 2; 3]. Cumulative errors may be increased with the increased degrees of freedom when using this method. POE (product of exponential) theory is another option for kinematics modeling [4; 5; 6; 7]. [PERSON] et al. gave a generalized solution of a kinematics problem based on POE in order to analyze the kinematics problems of serial robots [8]. [PERSON] et al. used POE theory to solve kinematics problems of industrial robots [9], POE theory allows global description of rigid body motion and greatly simplifies the analysis of mechanisms. For inverse kinematics, the common methods are the algebraic method, geometric method, and numerical method. [PERSON] et al. proposed a geometric method to find the optimal solution of the inverse kinematics of redundant or hyperredundant robots from infinite solutions [10]. [PERSON] et al. obtained the inverse kinematics solution by using iteration methods with the initial value of the analytical solution [11]. Modern intelligent algorithms are also widely used to solve the inverse kinematics of robots with complex structures. [PERSON] et al. used the adaptive fuzzy logic method to determine the inverse kinematics solution of a 3-DOF (degree of freedom) planar robot [12]. [PERSON] et al. obtained the solution of an inverse kinematics equation by using four different optimization algorithms [13]. These methods can produce accurate inverse kinematics solutions, but they need a lot of training. The inverse kinematics solution of robots is a multisolution problem. The criterion of shortest travel is generally adopted to select the appropriate solution. [PERSON] et al. proposed a criterion of weighted shortest travel, and selected the appropriate solution by finding the minimum value of weighted Euclidean distance [14]. [PERSON] et al. divided the result of inverse solution with the geometric attitude of the robot to facilitate selection [15]. The determination of the workspace boundary is generally an intermediate but critical step in analyzing multijoint industrial robots [16]. Traditionally, the workspace boundaries of multijoint industrial robots are usually expressed as numerical curves on the boundaries or analytical formulas of a certain kind of robot. [PERSON] et al. used differential geometry and topology methods to obtain general formulas for the workspace recognition and visualization using the manifold layering method [17]. [PERSON] et al. used the surface envelope overlay (SEO) method to identify and visualize the robot workspace [16]. Graphic methods are another way to find workspace boundaries [18]. [PERSON] et al. took advantage of the arm length of a robot and the angle ranges of its joints to obtain section screenshots of the robot's workspace on the xz plane via the geometric drawing method [19]. [PERSON] et al. discussed the geometric description of a robot's workspace and presented the design results of a delta robot for a given workspace [20]. This method is more intuitive, but it is not suitable for robots with more than 3 degrees of freedom.