# Algorithmic Modified Denavit-Hartenberg Modeling for Robotic Manipulators Using Line Geometry
## 5 Verification
### Ur5  
The UR-5 is one of the widely used six-dof robotic manipulators, which has six revolute joints. The left side of Figure 10 shows line information denoted by \((\hat{z}_{i},p_{i})\) including the base frame and TCF, where the line information was extracted from the \(\hat{z}\)-axis and the origin of the joint frame written in the URDF [32]. The right side of Figure 10 illustrates the coordinate frames defined by MDH parameters extracted from Algorithm 1. When the input data suggested in Table 13 are applied to Algorithm 1, the line geometry blocks are called by the algorithm, and the spatial relationships between the joint axes are determined as shown in Table 14.  
According to the above selection, the MDH can be obtained. As a result, a pair of input and output data for all joints are listed in Table 13 and denoted on the right side of Figure 10.  
Figure 10: MDH parameters of UR5, where the left side shows the input data regarding \(\hat{z}_{i}\) and \(p_{i}\) denoted by the circle \(\circ\), and the right side illustrates the frames obtained from the MDH parameters. Numerical data of input/output related to Algorithm 1 are given in Table 13.  
Figure 9: MDH parameters of KUKA LBR iiwa, where the left side shows the input data regarding \(\hat{z}_{i}\) and \(p_{i}\) denoted by the circle \(\circ\), and the right side illustrates the frames obtained from the MDH parameters. Numerical data of input/output related to Algorithm 1 are given in Table 11.  
of the proposed algorithm was verified through the conventional SCARA robot and three different commercial Indy7, KUKA LBR iiwa, and UR5 robots. Although the kinematic modeling using the MDH parameters was not new, the proposed algorithm was able to overcome the disadvantage that the MDH parameters are differently defined by user-defined coordinate frames, because the line geometry provides a unique and systematic method.  
Conceptualization, M.S.; Methodology, M.S.; Software, M.S.; Validation, M.S.; Formal analysis, M.S.; Investigation, M.S.; Data curation, M.S.; Writing--original draft, M.S.; Writing--review & editing, M.S. and Y.C.; Supervision, Y.C.; Project administration, Y.C.; Funding acquisition, Y.C. All authors have read and agreed to the published version of the manuscript.  
This work was supported in part by the Institute of Civil-Military Technology Cooperation funded by the Defense Acquisition Program Administration and the Ministry of Trade, Industry, and Energy of the Korean government under grant No. 22-CM-EC-36, and in part by the Industrial Strategic Technology Development Program funded by the Ministry of Trade, Industry, and Energy, under grant No. 20023257, South Korea.  
Not applicable.  
The source code and sample data used for the algorithmic MDH parameter extraction are openly available at [[https://github.com/Minchang5](https://github.com/Minchang5) ung0223/AlgorithmicMDH]([https://github.com/Minchang5](https://github.com/Minchang5) ung0223/AlgorithmicMDH) (accessed on 27 April 2025). Additional simulation results are available from the corresponding author upon reasonable request.  
The authors thank the organizations that provided data and simulation models used in this study.  
The authors declare no conflicts of interest. The sponsors had no role in the design, execution, interpretation, or writing of the study.