APPLIED MATHEMATICAL MODELING FOR 3D KINEMATIC SPATIAL RECONSTRUCTION IN A LOW-COST MONOCULAR WEBCAM-BASED SQUAT ANALYSIS SYSTEM

Abstract

Squat is a fundamental exercise for improving lower body strength; however, improper execution may increase the risk of musculoskeletal injury. Conventional motion analysis systems, such as marker-based technologies, provide high accuracy but require expensive equipment and controlled environments, while monocular camera-based approaches often suffer from limited three-dimensional representation. Therefore, this study proposes a low-cost squat analysis system, gui_mocap, which integrates monocular computer vision with vector-based mathematical modeling for real-time motion analysis. The system employs pose estimation to detect body landmarks and reconstructs joint kinematics in three-dimensional space using geometric vector operations. Knee joint angles are computed using the dot product formulation, and an Exponential Moving Average (EMA) filter is applied to improve measurement stability. Experimental evaluation was conducted using multiple squat repetitions to analyze motion patterns and consistency. The results demonstrate that the system can accurately identify key movement phases, including standing, deep flexion, and return to standing, while producing smooth and stable joint angle trajectories. Furthermore, the system is capable of analyzing repeated movements and generating descriptive statistics, such as average joint angles and range of motion, indicating consistent performance across repetitions. This study contributes a practical and affordable solution for real-time motion analysis using a monocular webcam, with potential applications in home-based exercise monitoring and basic rehabilitation.