Mathematical modelling and kinematic validation of a six-degree-of-freedom industrial manipulator usingthe robotics toolbox for Python

Abstract

 Accurate kinematic modeling is fundamental to the performance of industrial manipulators operating in precision-critical environments. This paper presents a forward kinematic analysis of a six-degree-of-freedom (6-DoF) serial industrial manipulator. The Denavit–Hartenberg (DH) convention is used to derive the homogeneous transformation matrices for each joint. The analytical model is implemented and validated using the open-source Robotics Toolbox for Python, with DH parameters sourced from RoboDK. End-effector poses computed by the toolbox are compared against manual DH matrix multiplication for multiple joint configurations, yielding zero numerical error across all tested poses and a maximum Jacobian error of $3.44 \times 10^{-7}$ (finite difference precision). The paper further extends the kinematic model to derive the geometric Jacobian matrix, perform singularity analysis via determinant and rank deficiency, compute Yoshikawa’s manipulability index, formulate a least-squares kinematic calibration framework, and characterize the reachable workspace using geometric and numerical methods. An extended mathematical treatment covering rigid body transformations, adjoint maps, twist and wrench representations, Hessian-based kinematic analysis, redundancy resolution, and operational-space dynamics is also presented. The close correspondence between analytical and numerical results validates the correctness of the derived DH parameters and demonstrates the effectiveness of open-source Python tools for industrial robot modeling and simulation.