# Algorithmic Modified Denavit-Hartenberg Modeling for Robotic Manipulators Using Line Geometry
## 1 Introduction  
Kinematic modeling of articulated robots is commonly categorized into two approaches according to the number of parameters required to describe the relative motion between adjacent joint axes: screw-based and line-based formulations. The screw-based formulation employs six parameters, whereas the line-based counterpart needs four. Owing to its theoretical completeness, the screw framework has been widely adopted in robotics research [1, 2, 3, 4]. In a screw-based model, the transformation of one link with respect to its predecessor is represented by an exponential map constructed from six exponential coordinates--three rotational and three linear components. Multiplying these exponential maps sequentially produces the forward kinematics of a single-branched manipulator and extends naturally to branched or closed-loop mechanisms. When each joint axis is regarded as an infinite line, only four independent quantities--Plucker coordinates--are sufficient to describe its pose, because a line is invariant under self-rotation and self-translation. Consequently, line-based formulations require four parameters per axis. Although screw coordinates offer superior numerical robustness for calibration, the modified Denavit-Hartenberg (MDH) notation remains the interface inindustrial robot controllers. Direct identification in the MDH, however, often suffers from convergence issues caused by the coupling inherent in its parameter structure [5]. To alleviate this limitation, the present study proposes a post-processing algorithm that converts screw-identified parameters into the MDH format. A concise comparison among the standard Product-of-Exponential (PoE) form, MDH notation, and the classical DH convention--including differences in parameter count, transformation dependencies, frame consistency, and automation suitability--is provided in Table 11 in Appendix A. Among traditional kinematic conventions, the Denavit-Hartenberg (DH) notation remains the most widely adopted. Originally introduced by [PERSON] and [PERSON], it expresses the relative transformation between successive frames with four scalar parameters [6; 7]. Two variants have since emerged: the distal form [8] and the proximal form--also called the Modified DH (MDH) notation--[1; 9; 10]. This work adopts the proximal form because every parameter is referenced to the same coordinate frame, which simplifies implementation and numerical treatment [11; 12; 13; 14]. Deriving MDH parameters, however, requires locating the common perpendicular between consecutive joint axes; line geometry therefore becomes an essential computational tool in our framework. The studies most closely related to this work are those of [PERSON] et al. [15] and [PERSON] et al. [16]. [PERSON] et al. extract line parameters by commanding axis-specific rotations and subsequently convert them to DH parameters, but their procedure requires user-defined joint-excitation trajectories for each axis during data acquisition and manual assignment of each frame's \(x\)-axis. [PERSON] et al. automate the extraction of conventional DH parameters, yet their method handles only revolute joints and is incompatible with the MDH formulation. The algorithm proposed in the present study overcomes all of these limitations: it requires no constrained excitation, supports both revolute and prismatic joints, and outputs parameters directly in the MDH notation. Recent studies [17; 18; 19; 20] have focused on kinematic parameter identification based on screw theory, typically performed with a robot equipped with sensors attached to a TCF for calibration. Although direct identification is not performed in this study, the proposed method can be subsequently applied to convert the identified screw-based parameters into an MDH representation. This allows for seamless integration with MDH-based kinematic modeling frameworks commonly used in industrial and simulation environments.