## 1 Introduction
Most of the analysis methods try to extract boundary information from singular points of robots by analyzing the characteristics of the singular points. Based on this theorem, several analytical criteria for determining the workspaces boundaries were derived [21]. [PERSON] et al. analyzed a workspace and subspace by polynomial criterion and Jacobian matrix, respectively [22; 23]. [PERSON] et al. described singular behavior with the row rank defect of the Jacobian matrix and presented the boundary subsurface of the envelope of a workspace using the perturbation method by dividing the singular surface into several subsurfaces [24]. Most numerical methods of singularities or bounds are mapped by grids. All grid-based methods produce an approximate boundary, and the accuracy depends on the size of the grid. The Monte Carlo method is commonly used to generate sample points in a grid in order to identify the grid points near the workspace boundary via the statistical method, which bypasses the necessity of tightness. [PERSON] et al. generated random values of the joint vector data by increasing the probability of the joint vector at the workspace boundary with the upper limit and lower limit of the joint angles [25]. [PERSON] et al. used the classical Monte Carlo method to generate an inaccurate workspace and used a Gaussian distribution to encrypt and extend the workspace until the boundary of the workspace was reached [26]. [PERSON] et al. added random points locally at the boundary points of each layer in order to generate a clear workspace boundary [27]. However, this method cannot exclude the identification of false holes in the workspace. The layering concept in 3D printing can be introduced to obtain expressions of workspace boundaries, which makes the visualization of the workspace feasible. In addition, interval-analysis-based methods have been used to analyze workspaces. Interval analysis can be used to evaluate the constraints, and branch-and-prune techniques used to characterize the constraint workspace. This is the basis of prescribed bounded velocity and force transmission factors. [PERSON] et al. proposed an interval-analysis-based approach to determine almost all the geometries of a simplified Gough platform for which the workspace included an arbitrary set of poses [28]. [PERSON] et al. used interval analysis to compute a dextrous workspace as well as the largest cube enclosed in this workspace [29]. [PERSON] et al. proposed a systematic algorithm based on the interval-analysis concept in order to find the maximal singularity-free circle or sphere within the workspace of parallel mechanisms [30]. [PERSON] et al. used interval analysis to evaluate several performance indexes of parallel kinematic machine design [31]. However, interval-analysis-based methods are mostly used to analyze the workspace of parallel mechanisms. This paper proposes a new workspace visualization method based on a 3D-printing layering concept in order to obtain and visualize the workspace accurately. At first, all possible solutions of joint angles are obtained in detail based on the forward kinematics and inverse kinematics of a multijoint industrial robot. Next, one multisolution selection method is proposed according to the sensitivity-based key degree of the joints to select the appropriate solutions. Next, a 3D-printing layering concept is introduced to establish one workspace visualization method. The forward kinematics can help to establish the boundary formula of each layer and obtain the limitation of the height of the workspace. The multisolution selection method can help to obtain the joint angles of each layer, which also can be used to print the workspace. The remains of the paper are organized as follows: In Section 2, the kinematics model of a multijoint industrial robot based on POE theory is established, and all possible joint-angle groups of one pose in the joint range are presented in detail. In Section 3, a multisolution selection method based on the key degree of joints is proposed.